43:
100:
2881:, viewed as a 1Ă1-matrix, which is nonsingular if and only if it is not zero. In this case, a non-degenerate critical point is a local maximum or a local minimum, depending on the sign of the second derivative, which is positive for a local minimum and negative for a local maximum. If the second derivative is null, the critical point is generally an
2938:
of a continuous function occur at critical points. Therefore, to find the local maxima and minima of a differentiable function, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the
Hessian matrix at these zeros. This requires the solution of a
2458:
3422:
This definition does not depend on the choice of the charts because the transitions maps being diffeomorphisms, their
Jacobian matrices are invertible and multiplying by them does not modify the rank of the Jacobian matrix of
2778:
A multiple root of the discriminant correspond either to several critical points or inflection asymptotes sharing the same critical value, or to a critical point which is also an inflection point, or to a singular point.
1244:, the critical point is the abscissa of the vertex, where the tangent line is horizontal, and the critical value is the ordinate of the vertex and may be represented by the intersection of this tangent line and the
1663:. The notion of critical point that is used in this section, may seem different from that of previous section. In fact it is the specialization to a simple case of the general notion of critical point given
2128:
670:
3026:
386:
1386:
3466:
3420:
3290:
543:
Although it is easily visualized on the graph (which is a curve), the notion of critical point of a function must not be confused with the notion of critical point, in some direction, of a
3334:
2726:
2629:
1013:
1589:
1905:
3585:
3070:
1848:
249:
3142:
2825:
1439:
1144:
498:
213:
1095:
930:
1299:
3375:
2346:
3205:
2753:
2684:
2575:
2548:
2490:
2327:
2300:
2248:
2165:
2042:
2001:
1956:
1794:
1767:
1715:
1532:
1234:
1482:
1188:
2209:
3612:
3676:
3656:
3636:
3553:
2465:
and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for
1623:
of the function. Thus for a polynomial function with only real roots, all critical points are real and are between the greatest and smallest roots.
17:
2877:
of the
Hessian determine the local behavior of the function. In the case of a function of a single variable, the Hessian is simply the
2054:
2931:
3685:
allows us to bound the number of connected components by a function of the degrees of the polynomials that define the variety.
615:
3954:
3893:
2978:
2854:. If the function is at least twice continuously differentiable the different cases may be distinguished by considering the
2947:
are much more efficient for finding local extrema, but cannot certify that all extrema have been found. In particular, in
3658:
contains at least a critical point, where the distance is minimal. It follows that the number of connected components of
333:
160:, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not not
152:, is a point in the domain of the function where the function derivative is equal to zero (or where the function is not
2962:, and modern algorithms for solving such systems provide competitive certified methods for finding the global minimum.
2788:
567:
167:
3426:
3380:
3921:
3845:
3812:
3734:
86:
64:
3254:
57:
1319:
3295:
2692:
2595:
2133:
This implies that this definition is a special case of the general definition of a critical point, which is given
3995:
31:
3535:
The link between critical points and topology already appears at a lower level of abstraction. For example, let
3761:
2959:
269:, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called
966:
1550:
4005:
407:
143:
1853:
1633:
in the complex plane, then there is at least one critical point within unit distance of any given root.
3785:
3558:
3043:
2755:
such the corresponding critical point is a point which is not singular nor an inflection point, or the
1799:
222:
3118:
2900:
of the critical point. A non-degenerate critical point is a local maximum if and only if the index is
2896:
variables, the number of negative eigenvalues of the
Hessian matrix at a critical point is called the
2801:
189:
4000:
3694:
2925:
2493:
2336:
1043:
883:
689:
391:
2453:{\displaystyle f(x,y)={\frac {\partial f}{\partial x}}(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}
3885:
3497:
3090:
is a called a critical value. A point in the complement of the set of critical values is called a
2909:
2905:
1740:
51:
1254:
3521:
3351:
3172:
2955:
2839:
is zero or undefined. The critical values are the values of the function at the critical points.
1724:
1608:
1404:
1100:
746:
529:
456:
445:
266:
153:
1626:
131:
is zero (or undefined, as specified below). The value of the function at a critical point is a
124:
68:
3682:
3178:
2731:
2662:
2553:
2526:
2468:
2305:
2278:
2226:
2143:
2020:
1979:
1934:
1772:
1745:
1679:
1492:
1201:
3881:
3075:
1736:
1452:
1158:
3873:
2179:
3840:. Edwards, Bruce H., 1946- (9th ed.). Belmont, Calif.: Brooks/Cole, Cengage Learning.
2339:
are considered as critical points. In fact the singular points are the points that satisfy
1728:
433:
161:
8:
3874:
2948:
2944:
2940:
2908:; it is a local minimum if the index is zero, or, equivalently, if the Hessian matrix is
2174:
2045:
1973:
544:
511:
3594:
3801:
3779:
3699:
3661:
3641:
3621:
3538:
3529:
2971:
2866:
2335:, but also on the choice of the coordinate axes. It depends also on the authors if the
1620:
1446:
181:
1024:
is a critical point with critical value 1 due to the derivative being equal to 0, and
3950:
3917:
3909:
3889:
3851:
3841:
3818:
3808:
3767:
3757:
3730:
3095:
2935:
2878:
1652:
725:
586:
284:
271:
3973:
3942:
3473:
2886:
2882:
2772:
537:
258:
157:
148:
108:
104:
3147:
3079:
2916:, that is a point which is a maximum in some directions and a minimum in others.
1908:
1732:
1656:
1642:
1031:
are critical points with critical value 0 due to the derivative being undefined.
262:
2795:(that is a set of values for the input variables, which is viewed as a point in
3249:
2859:
1487:
1191:
99:
3977:
3946:
1146:
This function has a unique critical point â1, because it is the unique number
3989:
3855:
3822:
3771:
2951:, these methods cannot certify that the output is really the global optimum.
2847:
2843:
1612:
1598:
is not a critical point because it is not included in the function's domain.
3525:
3099:
2913:
2851:
2632:
2517:
3681:
In the case of real algebraic varieties, this observation associated with
3245:
2912:. For the other values of the index, a non-degenerate critical point is a
3722:
1648:
1616:
1449:
at this point with vertical tangent. The corresponding critical value is
877:
280:
174:
116:
2842:
A critical point (where the function is differentiable) may be either a
3969:
2874:
2855:
449:
128:
1441:
otherwise, it is the unique critical point. The graph of the function
2762:
1630:
962:
If one considers the upper half circle as the graph of the function
867:
is a critical point of its graph for the projection parallel to the
800:
is a critical point of its graph for the projection parallel to the
3517:
3513:
2836:
2510:
is algebraic, that is when it is defined by a bivariate polynomial
2003:
of the critical point and of the tangent are the same point of the
1660:
1241:
171:
510:
of a critical point. These concepts may be visualized through the
265:
is not maximal. It extends further to differentiable maps between
2958:, the critical points and the critical values are solutions of a
1959:
522:
3756:. DemidovÇc, Boris P., Baranenkov, G. Moscow(IS): Moskva. 1964.
397:
2123:{\displaystyle f(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}
3638:
is a differential map such that each connected component of
672:
This means that the tangent of the curve is parallel to the
3098:
states that the set of critical values of a smooth map has
2134:
1664:
1544:
where it has a global minimum point, with critical value 0.
2965:
733:
varies, there are two branches of the curve on a side of
665:{\displaystyle {\tfrac {\partial g}{\partial y}}(x,y)=0.}
548:
1611:, all of a polynomial function's critical points in the
390:
In other words, the critical points are those where the
3021:{\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n},}
1647:
Critical points play an important role in the study of
1636:
3171:
These definitions extend to differential maps between
1629:
asserts that, if all of a function's roots lie in the
1570:
1534:
is differentiable everywhere except at critical point
1344:
620:
336:
170:, a critical point is a value in its domain where the
3664:
3644:
3624:
3597:
3561:
3541:
3429:
3383:
3354:
3298:
3257:
3181:
3158:. With this convention, all points are critical when
3121:
3105:
Some authors give a slightly different definition: a
3046:
2981:
2804:
2734:
2695:
2665:
2598:
2556:
2529:
2471:
2349:
2308:
2281:
2229:
2182:
2146:
2057:
2023:
1982:
1937:
1856:
1802:
1775:
1748:
1682:
1553:
1495:
1455:
1407:
1322:
1257:
1204:
1161:
1103:
1046:
969:
886:
618:
459:
225:
192:
3086:
is not maximal. The image of a critical point under
593:
of such a curve, for the projection parallel to the
381:{\textstyle {\frac {\partial f}{\partial y}}(x,y)=0}
3678:is bounded above by the number of critical points.
2889:, which may be a local minimum or a local maximum.
3807:(6th ed.). Belmont, CA: Thomson Brooks/Cole.
3800:
3670:
3650:
3630:
3606:
3579:
3547:
3460:
3414:
3369:
3328:
3284:
3199:
3136:
3064:
3020:
2819:
2747:
2720:
2678:
2623:
2569:
2542:
2484:
2452:
2321:
2294:
2242:
2203:
2159:
2122:
2036:
1995:
1950:
1899:
1842:
1788:
1761:
1709:
1583:
1526:
1476:
1433:
1380:
1293:
1228:
1182:
1138:
1097:is differentiable everywhere, with the derivative
1089:
1007:
924:
664:
492:
380:
243:
207:
3512:Critical points are fundamental for studying the
2520:is a useful tool to compute the critical points.
3987:
3461:{\displaystyle \psi \circ f\circ \varphi ^{-1}.}
3415:{\displaystyle \psi \circ f\circ \varphi ^{-1}.}
2865:A critical point at which the Hessian matrix is
1731:. The points of the curve are the points of the
521:at a critical point, the graph has a horizontal
27:Point where the derivative of a function is zero
3972:, Fundamentals of Differential Geometry p. 186,
304:the critical points of the projection onto the
3285:{\displaystyle \varphi :V\to \mathbb {R} ^{m}}
2919:
1602:
3872:Adams, Robert A.; Essex, Christopher (2009).
3524:. In particular, they are the basic tool for
2904:, or, equivalently, if the Hessian matrix is
2262:, and that the critical values are the same.
1739:satisfy the equation. There are two standard
1381:{\displaystyle f'(x)={\tfrac {2x^{-1/3}}{3}}}
3914:Differential geometry of curves and surfaces
3329:{\displaystyle \psi :W\to \mathbb {R} ^{n}.}
3207:be a differential map between two manifolds
2721:{\displaystyle \operatorname {Disc} _{y}(f)}
2653:. This discriminant is thus a polynomial in
2624:{\displaystyle \operatorname {Disc} _{y}(f)}
830:due to the tangent becoming parallel to the
398:Critical point of a single variable function
3480:is a real-valued function then we say that
2943:, which can be a difficult task. The usual
2275:as the points that are critical for either
3936:
3871:
3507:
2647:with coefficients that are polynomials in
2499:
680:does not define an implicit function from
3939:An Introduction to Differential Manifolds
3916:. Upper Saddle River, NJ: Prentice-Hall.
3564:
3476:(not necessarily finite dimensional) and
3313:
3272:
3124:
3049:
3005:
2990:
2807:
2771:-axis and is tangent "at infinity" to an
2044:if its coordinates are a solution of the
745:It follows from these definitions that a
723:. Such a critical point is also called a
228:
195:
103:The x-coordinates of the red circles are
87:Learn how and when to remove this message
3717:
3715:
876:For example, the critical points of the
98:
50:This article includes a list of general
3798:
2140:The definition of a critical point for
257:being, in this case, a point where the
14:
3988:
3835:
3721:
2966:Critical point of a differentiable map
1198:. The corresponding critical value is
1008:{\displaystyle f(x)={\sqrt {1-x^{2}}}}
3941:. Springer International Publishing.
3908:
3803:Calculus : early transcendentals
3712:
2523:Here we consider only the projection
142:More specifically, when dealing with
3867:
3865:
3748:
3746:
2782:
1637:Critical points of an implicit curve
1584:{\displaystyle f(x)={\tfrac {1}{x}}}
318:are the points where the tangent to
146:, a critical point, also known as a
36:
2954:When the function to minimize is a
2329:, although they depend not only on
940:for the projection parallel to the
180:This sort of definition extends to
24:
2789:function of several real variables
2420:
2412:
2382:
2374:
2090:
2082:
1900:{\displaystyle \pi _{x}((x,y))=y,}
1591:has no critical points. The point
955:for the direction parallel to the
631:
623:
348:
340:
168:function of several real variables
56:it lacks sufficient corresponding
25:
4017:
3862:
3754:Problems in mathematical analysis
3743:
3580:{\displaystyle \mathbb {R} ^{n},}
3223:. In the neighborhood of a point
3065:{\displaystyle \mathbb {R} ^{m},}
2689:More precisely, a simple root of
2659:which has the critical values of
2492:are exactly the points where the
1917:projection parallel to the x-axis
1913:projection parallel to the y-axis
1843:{\displaystyle \pi _{y}((x,y))=x}
612:), is a point of the curve where
244:{\displaystyle \mathbb {R} ^{n},}
177:is equal to zero (or undefined).
3137:{\displaystyle \mathbb {R} ^{m}}
2820:{\displaystyle \mathbb {R} ^{n}}
1676:defined by an implicit equation
676:-axis, and that, at this point,
208:{\displaystyle \mathbb {R} ^{m}}
156:). Similarly, when dealing with
41:
1090:{\displaystyle f(x)=x^{2}+2x+3}
925:{\displaystyle x^{2}+y^{2}-1=0}
710:is such a critical point, then
551:for a detailed definition). If
525:if one can be assigned at all.
18:Critical value (critical point)
3963:
3930:
3902:
3829:
3792:
3729:. Princeton University Press.
3614:The square of the distance to
3364:
3358:
3308:
3267:
3191:
3000:
2960:system of polynomial equations
2728:is either a critical value of
2715:
2709:
2618:
2612:
2441:
2429:
2403:
2391:
2365:
2353:
2198:
2192:
2111:
2099:
2073:
2061:
1968:exists and is parallel to the
1885:
1882:
1870:
1867:
1831:
1828:
1816:
1813:
1698:
1686:
1563:
1557:
1520:
1512:
1505:
1499:
1465:
1459:
1422:
1416:
1337:
1331:
1267:
1261:
1217:
1208:
1118:
1112:
1056:
1050:
979:
973:
653:
641:
481:
468:
369:
357:
13:
1:
3705:
1727:of two variables, commonly a
843:is again a critical point of
807:with the same critical value
3937:Lafontaine, Jacques (2015).
1907:that map the curve onto the
1294:{\displaystyle f(x)=x^{2/3}}
742:and zero on the other side.
144:functions of a real variable
7:
3910:Carmo, Manfredo PerdigĂŁo do
3876:Calculus: A Complete Course
3688:
3370:{\displaystyle \varphi (p)}
2920:Application to optimization
2835:if it is a point where the
2550:; Similar results apply to
1659:them and determining their
1603:Location of critical points
1434:{\displaystyle f'(x)\neq 0}
1139:{\displaystyle f'(x)=2x+2.}
1034:
493:{\displaystyle f'(x_{0})=0}
10:
4022:
3175:in the following way. Let
2923:
2641:viewed as a polynomial in
1670:Thus, we consider a curve
1640:
406:of a function of a single
29:
3978:10.1007/978-1-4612-0541-8
3947:10.1007/978-3-319-20735-3
3695:Singular point of a curve
3215:of respective dimensions
2926:Mathematical optimization
2765:which is parallel to the
2494:implicit function theorem
1972:-axis. In that case, the
1394:is not differentiable at
821:is not differentiable at
690:implicit function theorem
392:implicit function theorem
329:that is the points where
3522:real algebraic varieties
3200:{\displaystyle f:V\to W}
3173:differentiable manifolds
2748:{\displaystyle \pi _{y}}
2679:{\displaystyle \pi _{y}}
2570:{\displaystyle \pi _{x}}
2543:{\displaystyle \pi _{y}}
2485:{\displaystyle \pi _{y}}
2322:{\displaystyle \pi _{y}}
2295:{\displaystyle \pi _{x}}
2265:Some authors define the
2243:{\displaystyle \pi _{x}}
2160:{\displaystyle \pi _{x}}
2037:{\displaystyle \pi _{y}}
1996:{\displaystyle \pi _{y}}
1951:{\displaystyle \pi _{y}}
1789:{\displaystyle \pi _{x}}
1762:{\displaystyle \pi _{y}}
1710:{\displaystyle f(x,y)=0}
1527:{\displaystyle f(x)=|x|}
1229:{\displaystyle f(-1)=2.}
267:differentiable manifolds
3799:Stewart, James (2008).
3508:Application to topology
3484:is a critical point of
2956:multivariate polynomial
2873:, and the signs of the
2862:of second derivatives.
2775:(inflexion asymptote).
2500:Use of the discriminant
2256:is a critical point of
1725:differentiable function
1488:absolute value function
1477:{\displaystyle f(0)=0.}
1305:and differentiable for
1183:{\displaystyle 2x+2=0.}
747:differentiable function
570:of two variables, then
530:differentiable function
107:; the blue squares are
71:more precise citations.
3996:Multivariable calculus
3784:: CS1 maint: others (
3672:
3652:
3632:
3608:
3581:
3549:
3462:
3416:
3371:
3330:
3286:
3201:
3146:where the rank of the
3138:
3066:
3022:
2821:
2749:
2722:
2680:
2625:
2571:
2544:
2486:
2454:
2323:
2296:
2244:
2205:
2204:{\displaystyle y=g(x)}
2161:
2124:
2038:
1997:
1952:
1911:. They are called the
1901:
1844:
1790:
1763:
1711:
1585:
1528:
1478:
1435:
1382:
1295:
1230:
1184:
1140:
1091:
1009:
926:
729:, as, generally, when
666:
494:
382:
245:
209:
125:argument of a function
112:
3882:Pearson Prentice Hall
3673:
3653:
3633:
3609:
3582:
3555:be a sub-manifold of
3550:
3463:
3417:
3372:
3331:
3287:
3202:
3139:
3067:
3023:
2885:, but may also be an
2822:
2750:
2723:
2681:
2626:
2572:
2545:
2487:
2455:
2324:
2297:
2245:
2206:
2162:
2125:
2039:
1998:
1953:
1902:
1845:
1791:
1764:
1737:Cartesian coordinates
1712:
1586:
1529:
1479:
1436:
1383:
1296:
1231:
1185:
1141:
1092:
1010:
927:
759:has a critical point
719:is the corresponding
667:
495:
383:
246:
210:
102:
3836:Larson, Ron (2010).
3662:
3642:
3622:
3595:
3559:
3539:
3427:
3381:
3352:
3296:
3255:
3179:
3119:
3044:
2979:
2945:numerical algorithms
2941:system of equations
2802:
2732:
2693:
2663:
2596:
2554:
2527:
2469:
2347:
2306:
2279:
2227:
2180:
2144:
2055:
2021:
1980:
1935:
1854:
1800:
1773:
1746:
1729:bivariate polynomial
1680:
1655:, in particular for
1551:
1493:
1453:
1405:
1320:
1315:with the derivative
1255:
1202:
1159:
1101:
1044:
967:
884:
768:with critical value
616:
566:is a differentiable
457:
334:
322:are parallel to the
275:. In particular, if
223:
190:
30:For other uses, see
3591:be a point outside
2949:global optimization
2175:graph of a function
2046:system of equations
1627:Sendov's conjecture
1609:GaussâLucas theorem
1301:is defined for all
506:is the image under
182:differentiable maps
166:). Likewise, for a
127:where the function
4006:Singularity theory
3700:Singularity theory
3668:
3648:
3628:
3607:{\displaystyle V.}
3604:
3577:
3545:
3530:catastrophe theory
3458:
3412:
3367:
3326:
3282:
3197:
3134:
3062:
3038:are the points of
3018:
2972:differentiable map
2892:For a function of
2817:
2761:-coordinate of an
2745:
2718:
2676:
2621:
2567:
2540:
2482:
2450:
2319:
2292:
2240:
2201:
2157:
2120:
2034:
2011:. Thus a point of
2007:-axis, called the
1993:
1948:
1897:
1840:
1786:
1759:
1707:
1653:implicit equations
1581:
1579:
1524:
1474:
1431:
1378:
1376:
1291:
1226:
1180:
1136:
1087:
1005:
922:
662:
639:
528:Notice how, for a
490:
378:
272:bifurcation points
241:
205:
113:
3956:978-3-319-20734-6
3895:978-0-321-54928-0
3671:{\displaystyle V}
3651:{\displaystyle V}
3631:{\displaystyle V}
3548:{\displaystyle V}
2936:maxima and minima
2910:positive definite
2906:negative definite
2879:second derivative
2783:Several variables
2686:among its roots.
2427:
2389:
2097:
1578:
1375:
1003:
726:bifurcation point
638:
587:implicit equation
355:
285:implicit equation
158:complex variables
109:inflection points
105:stationary points
97:
96:
89:
16:(Redirected from
4013:
4001:Smooth functions
3980:
3967:
3961:
3960:
3934:
3928:
3927:
3906:
3900:
3899:
3879:
3869:
3860:
3859:
3833:
3827:
3826:
3806:
3796:
3790:
3789:
3783:
3775:
3750:
3741:
3740:
3719:
3683:BĂ©zout's theorem
3677:
3675:
3674:
3669:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3617:
3613:
3611:
3610:
3605:
3590:
3586:
3584:
3583:
3578:
3573:
3572:
3567:
3554:
3552:
3551:
3546:
3503:
3491:
3487:
3483:
3479:
3474:Hilbert manifold
3471:
3467:
3465:
3464:
3459:
3454:
3453:
3421:
3419:
3418:
3413:
3408:
3407:
3377:is critical for
3376:
3374:
3373:
3368:
3347:
3339:
3335:
3333:
3332:
3327:
3322:
3321:
3316:
3291:
3289:
3288:
3283:
3281:
3280:
3275:
3243:
3232:
3226:
3222:
3218:
3214:
3210:
3206:
3204:
3203:
3198:
3167:
3157:
3153:
3145:
3143:
3141:
3140:
3135:
3133:
3132:
3127:
3112:
3089:
3085:
3073:
3071:
3069:
3068:
3063:
3058:
3057:
3052:
3037:
3029:
3027:
3025:
3024:
3019:
3014:
3013:
3008:
2999:
2998:
2993:
2932:Fermat's theorem
2903:
2895:
2887:undulation point
2883:inflection point
2830:
2828:
2826:
2824:
2823:
2818:
2816:
2815:
2810:
2794:
2773:inflection point
2770:
2760:
2754:
2752:
2751:
2746:
2744:
2743:
2727:
2725:
2724:
2719:
2705:
2704:
2685:
2683:
2682:
2677:
2675:
2674:
2658:
2652:
2646:
2640:
2630:
2628:
2627:
2622:
2608:
2607:
2588:
2582:
2576:
2574:
2573:
2568:
2566:
2565:
2549:
2547:
2546:
2541:
2539:
2538:
2515:
2509:
2496:does not apply.
2491:
2489:
2488:
2483:
2481:
2480:
2461:
2459:
2457:
2456:
2451:
2428:
2426:
2418:
2410:
2390:
2388:
2380:
2372:
2334:
2328:
2326:
2325:
2320:
2318:
2317:
2301:
2299:
2298:
2293:
2291:
2290:
2274:
2261:
2255:
2249:
2247:
2246:
2241:
2239:
2238:
2223:is critical for
2222:
2210:
2208:
2207:
2202:
2172:
2166:
2164:
2163:
2158:
2156:
2155:
2129:
2127:
2126:
2121:
2098:
2096:
2088:
2080:
2043:
2041:
2040:
2035:
2033:
2032:
2017:is critical for
2016:
2002:
2000:
1999:
1994:
1992:
1991:
1967:
1957:
1955:
1954:
1949:
1947:
1946:
1927:
1919:, respectively.
1906:
1904:
1903:
1898:
1866:
1865:
1849:
1847:
1846:
1841:
1812:
1811:
1795:
1793:
1792:
1787:
1785:
1784:
1768:
1766:
1765:
1760:
1758:
1757:
1722:
1716:
1714:
1713:
1708:
1675:
1597:
1590:
1588:
1587:
1582:
1580:
1571:
1543:
1541:
1533:
1531:
1530:
1525:
1523:
1515:
1483:
1481:
1480:
1475:
1444:
1440:
1438:
1437:
1432:
1415:
1400:
1393:
1389:
1387:
1385:
1384:
1379:
1377:
1371:
1370:
1369:
1365:
1345:
1330:
1314:
1312:
1304:
1300:
1298:
1297:
1292:
1290:
1289:
1285:
1247:
1240:is a concave up
1239:
1235:
1233:
1232:
1227:
1197:
1190:This point is a
1189:
1187:
1186:
1181:
1154:
1145:
1143:
1142:
1137:
1111:
1096:
1094:
1093:
1088:
1071:
1070:
1030:
1023:
1016:
1014:
1012:
1011:
1006:
1004:
1002:
1001:
986:
961:
959:
954:
950:
946:
944:
939:
935:
931:
929:
928:
923:
909:
908:
896:
895:
873:
871:
866:
865:
846:
842:
833:
829:
820:
816:
806:
804:
799:
798:
779:
777:
767:
758:
741:
732:
718:
709:
687:
683:
675:
671:
669:
668:
663:
640:
637:
629:
621:
611:
596:
584:
565:
538:stationary point
520:
518:
509:
501:
499:
497:
496:
491:
480:
479:
467:
443:
439:
431:
430:
420:
394:does not apply.
389:
387:
385:
384:
379:
356:
354:
346:
338:
328:
326:
321:
317:
315:
311:parallel to the
310:
308:
303:
301:
283:, defined by an
278:
252:
250:
248:
247:
242:
237:
236:
231:
216:
214:
212:
211:
206:
204:
203:
198:
149:stationary point
137:
136:
92:
85:
81:
78:
72:
67:this article by
58:inline citations
45:
44:
37:
21:
4021:
4020:
4016:
4015:
4014:
4012:
4011:
4010:
3986:
3985:
3984:
3983:
3968:
3964:
3957:
3935:
3931:
3924:
3907:
3903:
3896:
3870:
3863:
3848:
3834:
3830:
3815:
3797:
3793:
3777:
3776:
3764:
3752:
3751:
3744:
3737:
3720:
3713:
3708:
3691:
3663:
3660:
3659:
3643:
3640:
3639:
3623:
3620:
3619:
3615:
3596:
3593:
3592:
3588:
3568:
3563:
3562:
3560:
3557:
3556:
3540:
3537:
3536:
3510:
3501:
3489:
3485:
3481:
3477:
3469:
3446:
3442:
3428:
3425:
3424:
3400:
3396:
3382:
3379:
3378:
3353:
3350:
3349:
3345:
3337:
3317:
3312:
3311:
3297:
3294:
3293:
3276:
3271:
3270:
3256:
3253:
3252:
3250:diffeomorphisms
3234:
3228:
3224:
3220:
3216:
3212:
3208:
3180:
3177:
3176:
3159:
3155:
3151:
3148:Jacobian matrix
3128:
3123:
3122:
3120:
3117:
3116:
3114:
3110:
3087:
3083:
3080:Jacobian matrix
3053:
3048:
3047:
3045:
3042:
3041:
3039:
3035:
3032:critical points
3009:
3004:
3003:
2994:
2989:
2988:
2980:
2977:
2976:
2974:
2968:
2928:
2922:
2901:
2893:
2811:
2806:
2805:
2803:
2800:
2799:
2797:
2796:
2792:
2785:
2766:
2756:
2739:
2735:
2733:
2730:
2729:
2700:
2696:
2694:
2691:
2690:
2670:
2666:
2664:
2661:
2660:
2654:
2648:
2642:
2636:
2603:
2599:
2597:
2594:
2593:
2584:
2578:
2561:
2557:
2555:
2552:
2551:
2534:
2530:
2528:
2525:
2524:
2511:
2505:
2504:When the curve
2502:
2476:
2472:
2470:
2467:
2466:
2419:
2411:
2409:
2381:
2373:
2371:
2348:
2345:
2344:
2343:
2337:singular points
2330:
2313:
2309:
2307:
2304:
2303:
2286:
2282:
2280:
2277:
2276:
2270:
2267:critical points
2257:
2251:
2250:if and only if
2234:
2230:
2228:
2225:
2224:
2212:
2181:
2178:
2177:
2168:
2167:is similar. If
2151:
2147:
2145:
2142:
2141:
2089:
2081:
2079:
2056:
2053:
2052:
2028:
2024:
2022:
2019:
2018:
2012:
1987:
1983:
1981:
1978:
1977:
1963:
1942:
1938:
1936:
1933:
1932:
1923:
1909:coordinate axes
1861:
1857:
1855:
1852:
1851:
1807:
1803:
1801:
1798:
1797:
1780:
1776:
1774:
1771:
1770:
1753:
1749:
1747:
1744:
1743:
1733:Euclidean plane
1718:
1681:
1678:
1677:
1671:
1645:
1643:Algebraic curve
1639:
1615:are within the
1605:
1592:
1569:
1552:
1549:
1548:
1536:
1535:
1519:
1511:
1494:
1491:
1490:
1454:
1451:
1450:
1442:
1408:
1406:
1403:
1402:
1395:
1391:
1361:
1354:
1350:
1346:
1343:
1323:
1321:
1318:
1317:
1316:
1307:
1306:
1302:
1281:
1277:
1273:
1256:
1253:
1252:
1245:
1237:
1203:
1200:
1199:
1195:
1160:
1157:
1156:
1153:
1147:
1104:
1102:
1099:
1098:
1066:
1062:
1045:
1042:
1041:
1037:
1025:
1018:
997:
993:
985:
968:
965:
964:
963:
957:
956:
952:
948:
942:
941:
937:
933:
904:
900:
891:
887:
885:
882:
881:
869:
868:
863:
856:
849:
848:
844:
841:
835:
831:
828:
822:
818:
813:
808:
802:
801:
796:
789:
782:
781:
780:if and only if
776:
770:
769:
766:
760:
749:
740:
734:
730:
717:
711:
707:
700:
693:
685:
681:
673:
630:
622:
619:
617:
614:
613:
598:
597:-axis (the map
594:
571:
552:
536:is the same as
516:
515:
507:
475:
471:
460:
458:
455:
454:
453:
441:
437:
429:
423:
422:
411:
400:
347:
339:
337:
335:
332:
331:
330:
324:
323:
319:
313:
312:
306:
305:
288:
287:
276:
263:Jacobian matrix
232:
227:
226:
224:
221:
220:
218:
199:
194:
193:
191:
188:
187:
185:
134:
133:
93:
82:
76:
73:
63:Please help to
62:
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
4019:
4009:
4008:
4003:
3998:
3982:
3981:
3962:
3955:
3929:
3922:
3901:
3894:
3861:
3846:
3828:
3813:
3791:
3762:
3742:
3735:
3710:
3709:
3707:
3704:
3703:
3702:
3697:
3690:
3687:
3667:
3647:
3627:
3618:of a point of
3603:
3600:
3576:
3571:
3566:
3544:
3509:
3506:
3457:
3452:
3449:
3445:
3441:
3438:
3435:
3432:
3411:
3406:
3403:
3399:
3395:
3392:
3389:
3386:
3366:
3363:
3360:
3357:
3325:
3320:
3315:
3310:
3307:
3304:
3301:
3279:
3274:
3269:
3266:
3263:
3260:
3196:
3193:
3190:
3187:
3184:
3131:
3126:
3113:is a point of
3107:critical point
3096:Sard's theorem
3061:
3056:
3051:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2984:
2967:
2964:
2924:Main article:
2921:
2918:
2869:is said to be
2860:Hessian matrix
2814:
2809:
2784:
2781:
2742:
2738:
2717:
2714:
2711:
2708:
2703:
2699:
2673:
2669:
2620:
2617:
2614:
2611:
2606:
2602:
2577:by exchanging
2564:
2560:
2537:
2533:
2501:
2498:
2479:
2475:
2463:
2462:
2449:
2446:
2443:
2440:
2437:
2434:
2431:
2425:
2422:
2417:
2414:
2408:
2405:
2402:
2399:
2396:
2393:
2387:
2384:
2379:
2376:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2316:
2312:
2289:
2285:
2237:
2233:
2200:
2197:
2194:
2191:
2188:
2185:
2154:
2150:
2131:
2130:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2095:
2092:
2087:
2084:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2031:
2027:
2009:critical value
1990:
1986:
1945:
1941:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1864:
1860:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1810:
1806:
1783:
1779:
1756:
1752:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1638:
1635:
1604:
1601:
1600:
1599:
1577:
1574:
1568:
1565:
1562:
1559:
1556:
1545:
1522:
1518:
1514:
1510:
1507:
1504:
1501:
1498:
1484:
1473:
1470:
1467:
1464:
1461:
1458:
1430:
1427:
1424:
1421:
1418:
1414:
1411:
1374:
1368:
1364:
1360:
1357:
1353:
1349:
1342:
1339:
1336:
1333:
1329:
1326:
1288:
1284:
1280:
1276:
1272:
1269:
1266:
1263:
1260:
1249:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1192:global minimum
1179:
1176:
1173:
1170:
1167:
1164:
1151:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1110:
1107:
1086:
1083:
1080:
1077:
1074:
1069:
1065:
1061:
1058:
1055:
1052:
1049:
1036:
1033:
1000:
996:
992:
989:
984:
981:
978:
975:
972:
921:
918:
915:
912:
907:
903:
899:
894:
890:
861:
854:
839:
826:
811:
794:
787:
774:
764:
738:
721:critical value
715:
705:
698:
661:
658:
655:
652:
649:
646:
643:
636:
633:
628:
625:
591:critical point
589:of a curve. A
534:critical point
504:critical value
489:
486:
483:
478:
474:
470:
466:
463:
446:differentiable
427:
404:critical point
399:
396:
377:
374:
371:
368:
365:
362:
359:
353:
350:
345:
342:
255:critical point
240:
235:
230:
202:
197:
154:differentiable
135:critical value
121:critical point
95:
94:
49:
47:
40:
32:Critical point
26:
9:
6:
4:
3:
2:
4018:
4007:
4004:
4002:
3999:
3997:
3994:
3993:
3991:
3979:
3975:
3971:
3966:
3958:
3952:
3948:
3944:
3940:
3933:
3925:
3923:0-13-212589-7
3919:
3915:
3911:
3905:
3897:
3891:
3887:
3883:
3878:
3877:
3868:
3866:
3857:
3853:
3849:
3847:9780547167022
3843:
3839:
3832:
3824:
3820:
3816:
3814:9780495011668
3810:
3805:
3804:
3795:
3787:
3781:
3773:
3769:
3765:
3759:
3755:
3749:
3747:
3738:
3736:0-691-08008-9
3732:
3728:
3724:
3718:
3716:
3711:
3701:
3698:
3696:
3693:
3692:
3686:
3684:
3679:
3665:
3645:
3625:
3601:
3598:
3574:
3569:
3542:
3533:
3531:
3527:
3523:
3519:
3515:
3505:
3499:
3495:
3475:
3455:
3450:
3447:
3443:
3439:
3436:
3433:
3430:
3409:
3404:
3401:
3397:
3393:
3390:
3387:
3384:
3361:
3355:
3343:
3323:
3318:
3305:
3302:
3299:
3277:
3264:
3261:
3258:
3251:
3247:
3241:
3237:
3231:
3194:
3188:
3185:
3182:
3174:
3169:
3166:
3162:
3154:is less than
3149:
3129:
3108:
3103:
3101:
3097:
3093:
3092:regular value
3081:
3077:
3059:
3054:
3033:
3015:
3010:
2995:
2985:
2982:
2973:
2963:
2961:
2957:
2952:
2950:
2946:
2942:
2937:
2933:
2927:
2917:
2915:
2911:
2907:
2899:
2890:
2888:
2884:
2880:
2876:
2872:
2871:nondegenerate
2868:
2863:
2861:
2857:
2853:
2849:
2848:local minimum
2845:
2844:local maximum
2840:
2838:
2834:
2812:
2790:
2780:
2776:
2774:
2769:
2764:
2759:
2740:
2736:
2712:
2706:
2701:
2697:
2687:
2671:
2667:
2657:
2651:
2645:
2639:
2634:
2615:
2609:
2604:
2600:
2590:
2587:
2581:
2562:
2558:
2535:
2531:
2521:
2519:
2514:
2508:
2497:
2495:
2477:
2473:
2447:
2444:
2438:
2435:
2432:
2423:
2415:
2406:
2400:
2397:
2394:
2385:
2377:
2368:
2362:
2359:
2356:
2350:
2342:
2341:
2340:
2338:
2333:
2314:
2310:
2287:
2283:
2273:
2268:
2263:
2260:
2254:
2235:
2231:
2220:
2216:
2195:
2189:
2186:
2183:
2176:
2171:
2152:
2148:
2138:
2136:
2117:
2114:
2108:
2105:
2102:
2093:
2085:
2076:
2070:
2067:
2064:
2058:
2051:
2050:
2049:
2047:
2029:
2025:
2015:
2010:
2006:
1988:
1984:
1975:
1971:
1966:
1961:
1943:
1939:
1931:
1926:
1920:
1918:
1914:
1910:
1894:
1891:
1888:
1879:
1876:
1873:
1862:
1858:
1837:
1834:
1825:
1822:
1819:
1808:
1804:
1796:, defined by
1781:
1777:
1754:
1750:
1742:
1738:
1734:
1730:
1726:
1721:
1704:
1701:
1695:
1692:
1689:
1683:
1674:
1668:
1666:
1662:
1658:
1654:
1650:
1644:
1634:
1632:
1628:
1624:
1622:
1618:
1614:
1613:complex plane
1610:
1595:
1575:
1572:
1566:
1560:
1554:
1547:The function
1546:
1539:
1516:
1508:
1502:
1496:
1489:
1485:
1471:
1468:
1462:
1456:
1448:
1428:
1425:
1419:
1412:
1409:
1398:
1372:
1366:
1362:
1358:
1355:
1351:
1347:
1340:
1334:
1327:
1324:
1310:
1286:
1282:
1278:
1274:
1270:
1264:
1258:
1251:The function
1250:
1243:
1236:The graph of
1223:
1220:
1214:
1211:
1205:
1193:
1177:
1174:
1171:
1168:
1165:
1162:
1150:
1133:
1130:
1127:
1124:
1121:
1115:
1108:
1105:
1084:
1081:
1078:
1075:
1072:
1067:
1063:
1059:
1053:
1047:
1040:The function
1039:
1038:
1032:
1028:
1021:
998:
994:
990:
987:
982:
976:
970:
919:
916:
913:
910:
905:
901:
897:
892:
888:
879:
874:
860:
853:
838:
825:
815:
793:
786:
773:
763:
756:
752:
748:
743:
737:
728:
727:
722:
714:
704:
697:
691:
679:
659:
656:
650:
647:
644:
634:
626:
610:
606:
602:
592:
588:
582:
578:
574:
569:
563:
559:
555:
550:
546:
541:
539:
535:
531:
526:
524:
513:
505:
487:
484:
476:
472:
464:
461:
451:
447:
435:
426:
421:, is a value
418:
414:
409:
408:real variable
405:
395:
393:
375:
372:
366:
363:
360:
351:
343:
299:
295:
291:
286:
282:
274:
273:
268:
264:
260:
256:
238:
233:
200:
183:
178:
176:
173:
169:
165:
164:
159:
155:
151:
150:
145:
140:
138:
130:
126:
122:
118:
110:
106:
101:
91:
88:
80:
70:
66:
60:
59:
53:
48:
39:
38:
33:
19:
3965:
3938:
3932:
3913:
3904:
3875:
3837:
3831:
3802:
3794:
3753:
3727:Morse Theory
3726:
3723:Milnor, John
3680:
3534:
3526:Morse theory
3511:
3493:
3341:
3239:
3235:
3229:
3170:
3164:
3160:
3106:
3104:
3100:measure zero
3091:
3031:
2969:
2953:
2934:, all local
2929:
2914:saddle point
2897:
2891:
2870:
2864:
2852:saddle point
2841:
2832:
2786:
2777:
2767:
2757:
2688:
2655:
2649:
2643:
2637:
2633:discriminant
2591:
2585:
2579:
2522:
2518:discriminant
2512:
2506:
2503:
2464:
2331:
2271:
2266:
2264:
2258:
2252:
2218:
2214:
2169:
2139:
2132:
2013:
2008:
2004:
1969:
1964:
1930:critical for
1929:
1924:
1921:
1916:
1912:
1719:
1672:
1669:
1649:plane curves
1646:
1625:
1606:
1593:
1537:
1396:
1308:
1148:
1026:
1019:
880:of equation
875:
858:
851:
836:
834:-axis, then
823:
809:
791:
784:
771:
761:
754:
750:
744:
735:
724:
720:
712:
702:
695:
677:
608:
604:
600:
590:
580:
576:
572:
561:
557:
553:
542:
533:
527:
503:
424:
416:
412:
403:
401:
297:
293:
289:
270:
254:
179:
162:
147:
141:
132:
120:
114:
83:
77:January 2015
74:
55:
2875:eigenvalues
2867:nonsingular
2856:eigenvalues
2516:, then the
1922:A point of
1741:projections
1651:defined by
1617:convex hull
878:unit circle
452:is 0 (i.e.
281:plane curve
163:holomorphic
117:mathematics
69:introducing
3990:Categories
3970:Serge Lang
3884:. p.
3763:0846407612
3706:References
3498:submersion
3336:The point
3074:where the
2791:, a point
1641:See also:
1155:for which
847:, but now
450:derivative
129:derivative
52:references
3856:319729593
3823:144526840
3780:cite book
3772:799468131
3518:manifolds
3448:−
3444:φ
3440:∘
3434:∘
3431:ψ
3402:−
3398:φ
3394:∘
3388:∘
3385:ψ
3356:φ
3309:→
3300:ψ
3268:→
3259:φ
3192:→
3001:→
2763:asymptote
2737:π
2707:
2668:π
2610:
2559:π
2532:π
2474:π
2421:∂
2413:∂
2383:∂
2375:∂
2311:π
2284:π
2232:π
2149:π
2091:∂
2083:∂
2026:π
1985:π
1958:, if the
1940:π
1859:π
1805:π
1778:π
1751:π
1657:sketching
1631:unit disk
1426:≠
1356:−
1212:−
991:−
911:−
632:∂
624:∂
349:∂
341:∂
3912:(1976).
3838:Calculus
3725:(1963).
3689:See also
3514:topology
3342:critical
2970:Given a
2837:gradient
2833:critical
1915:and the
1717:, where
1661:topology
1413:′
1328:′
1242:parabola
1109:′
1035:Examples
568:function
465:′
184:between
172:gradient
3233:and of
3144:
3115:
3078:of the
3072:
3040:
3028:
2975:
2858:of the
2827:
2798:
2631:be the
2211:, then
2173:is the
1960:tangent
1619:of the
1607:By the
953:(-1, 0)
938:(0, -1)
585:is the
523:tangent
448:or its
444:is not
432:in the
261:of the
251:
219:
215:
186:
123:is the
65:improve
3953:
3920:
3892:
3854:
3844:
3821:
3811:
3770:
3760:
3733:
3246:charts
2787:For a
1974:images
1735:whose
1445:has a
1390:Since
1248:-axis.
960:-axis.
949:(1, 0)
945:-axis,
934:(0, 1)
872:-axis.
805:-axis,
692:). If
440:where
434:domain
327:-axis,
309:-axis,
54:, but
3472:is a
3163:<
2898:index
2850:or a
2135:below
1723:is a
1665:below
1621:roots
1017:then
688:(see
583:) = 0
549:below
547:(see
545:curve
512:graph
316:-axis
300:) = 0
279:is a
3951:ISBN
3918:ISBN
3890:ISBN
3852:OCLC
3842:ISBN
3819:OCLC
3809:ISBN
3786:link
3768:OCLC
3758:ISBN
3731:ISBN
3587:and
3528:and
3520:and
3344:for
3292:and
3248:are
3219:and
3211:and
3076:rank
3030:the
2846:, a
2698:Disc
2601:Disc
2592:Let
2583:and
1850:and
1769:and
1486:The
1447:cusp
1401:and
1029:= ±1
951:and
947:and
936:and
932:are
607:) â
259:rank
217:and
175:norm
119:, a
3974:doi
3943:doi
3886:744
3516:of
3500:at
3494:not
3492:is
3488:if
3468:If
3348:if
3340:is
3227:of
3150:of
3109:of
3082:of
3034:of
2930:By
2831:is
2635:of
2302:or
2269:of
1976:by
1962:to
1928:is
1596:= 0
1540:= 0
1399:= 0
1311:â 0
1194:of
1022:= 0
817:If
684:to
514:of
436:of
115:In
3992::
3949:.
3888:.
3880:.
3864:^
3850:.
3817:.
3782:}}
3778:{{
3766:.
3745:^
3714:^
3532:.
3504:.
3496:a
3244:,
3168:.
3102:.
3094:.
2589:.
2217:,
2137:.
2048::
1667:.
1472:0.
1224:2.
1178:0.
1134:2.
857:,
790:,
701:,
660:0.
603:,
560:,
540:.
532:,
502:A
500:).
410:,
402:A
253:a
139:.
3976::
3959:.
3945::
3926:.
3898:.
3858:.
3825:.
3788:)
3774:.
3739:.
3666:V
3646:V
3626:V
3616:P
3602:.
3599:V
3589:P
3575:,
3570:n
3565:R
3543:V
3502:p
3490:f
3486:f
3482:p
3478:f
3470:M
3456:.
3451:1
3437:f
3410:.
3405:1
3391:f
3365:)
3362:p
3359:(
3346:f
3338:p
3324:.
3319:n
3314:R
3306:W
3303::
3278:m
3273:R
3265:V
3262::
3242:)
3240:p
3238:(
3236:f
3230:V
3225:p
3221:n
3217:m
3213:W
3209:V
3195:W
3189:V
3186::
3183:f
3165:n
3161:m
3156:n
3152:f
3130:m
3125:R
3111:f
3088:f
3084:f
3060:,
3055:m
3050:R
3036:f
3016:,
3011:n
3006:R
2996:m
2991:R
2986::
2983:f
2902:n
2894:n
2829:)
2813:n
2808:R
2793:P
2768:y
2758:x
2741:y
2716:)
2713:f
2710:(
2702:y
2672:y
2656:x
2650:x
2644:y
2638:f
2619:)
2616:f
2613:(
2605:y
2586:y
2580:x
2563:x
2536:y
2513:f
2507:C
2478:y
2460:,
2448:0
2445:=
2442:)
2439:y
2436:,
2433:x
2430:(
2424:y
2416:f
2407:=
2404:)
2401:y
2398:,
2395:x
2392:(
2386:x
2378:f
2369:=
2366:)
2363:y
2360:,
2357:x
2354:(
2351:f
2332:C
2315:y
2288:x
2272:C
2259:g
2253:x
2236:x
2221:)
2219:y
2215:x
2213:(
2199:)
2196:x
2193:(
2190:g
2187:=
2184:y
2170:C
2153:x
2118:0
2115:=
2112:)
2109:y
2106:,
2103:x
2100:(
2094:y
2086:f
2077:=
2074:)
2071:y
2068:,
2065:x
2062:(
2059:f
2030:y
2014:C
2005:x
1989:y
1970:y
1965:C
1944:y
1925:C
1895:,
1892:y
1889:=
1886:)
1883:)
1880:y
1877:,
1874:x
1871:(
1868:(
1863:x
1838:x
1835:=
1832:)
1829:)
1826:y
1823:,
1820:x
1817:(
1814:(
1809:y
1782:x
1755:y
1720:f
1705:0
1702:=
1699:)
1696:y
1693:,
1690:x
1687:(
1684:f
1673:C
1594:x
1576:x
1573:1
1567:=
1564:)
1561:x
1558:(
1555:f
1542:,
1538:x
1521:|
1517:x
1513:|
1509:=
1506:)
1503:x
1500:(
1497:f
1469:=
1466:)
1463:0
1460:(
1457:f
1443:f
1429:0
1423:)
1420:x
1417:(
1410:f
1397:x
1392:f
1388:.
1373:3
1367:3
1363:/
1359:1
1352:x
1348:2
1341:=
1338:)
1335:x
1332:(
1325:f
1313:,
1309:x
1303:x
1287:3
1283:/
1279:2
1275:x
1271:=
1268:)
1265:x
1262:(
1259:f
1246:y
1238:f
1221:=
1218:)
1215:1
1209:(
1206:f
1196:f
1175:=
1172:2
1169:+
1166:x
1163:2
1152:0
1149:x
1131:+
1128:x
1125:2
1122:=
1119:)
1116:x
1113:(
1106:f
1085:3
1082:+
1079:x
1076:2
1073:+
1068:2
1064:x
1060:=
1057:)
1054:x
1051:(
1048:f
1027:x
1020:x
1015:,
999:2
995:x
988:1
983:=
980:)
977:x
974:(
971:f
958:y
943:x
920:0
917:=
914:1
906:2
902:y
898:+
893:2
889:x
870:y
864:)
862:0
859:y
855:0
852:x
850:(
845:f
840:0
837:x
832:y
827:0
824:x
819:f
814:.
812:0
810:y
803:x
797:)
795:0
792:y
788:0
785:x
783:(
778:,
775:0
772:y
765:0
762:x
757:)
755:x
753:(
751:f
739:0
736:x
731:x
716:0
713:x
708:)
706:0
703:y
699:0
696:x
694:(
686:y
682:x
678:g
674:y
657:=
654:)
651:y
648:,
645:x
642:(
635:y
627:g
609:x
605:y
601:x
599:(
595:y
581:y
579:,
577:x
575:(
573:g
564:)
562:y
558:x
556:(
554:g
519::
517:f
508:f
488:0
485:=
482:)
477:0
473:x
469:(
462:f
442:f
438:f
428:0
425:x
419:)
417:x
415:(
413:f
388:.
376:0
373:=
370:)
367:y
364:,
361:x
358:(
352:y
344:f
325:y
320:C
314:y
307:x
302:,
298:y
296:,
294:x
292:(
290:f
277:C
239:,
234:n
229:R
201:m
196:R
111:.
90:)
84:(
79:)
75:(
61:.
34:.
20:)
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