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Only Critical Point in Town Test

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PartialDerivative

The only critical point in town test states that a univariate real function f(x) with a single critical point that is a local maximum attains its global maximum there (Wagon 1991, p. 87). The test breaks down for bivariate functions, but does hold for bivariate polynomials of degree <=4. Such exceptions include

z=3xe^y-x^3-e^(3y)
(1)
z=x^2(1+y)^3+y^2
(2)

(Rosenholtz and Smylie 1985, Wagon 1991).

A related example concerning mixed partial derivatives is

z(x,y)={(xy(x^2-y^2))/(x^2+y^2) for (x,y)!=(0,0); 0 for (x,y)=(0,0).
(3)

This function has a saddle point at the origin, since its values have opposite signs along y=x/2 and y=2x. Direct differentiation gives the unequal mixed partial derivatives z_(yx)(0,0)=1 and z_(xy)(0,0)=-1. Both mixed partial derivatives are discontinuous at the origin.

For polynomials in three variables, the quartic

p(x,y,z)=y^2+(1+y)z^2+(z-x(1+y))^2
(4)

has a unique critical point at the origin, where it has a nondegenerate strict local minimum (Aduén 2026). Its quadratic part y^2+z^2+(z-x)^2 is a positive definite quadratic form, but

p(t,-2,-t)=4-t^2->-infty
(5)

as |t|->infty, so this local minimum is not global. To check uniqueness, put h=z-x(1+y). The equations for the gradient are

p_x=-2(1+y)h
(6)
p_y=2y+z^2-2xh
(7)
p_z=2(1+y)z+2h.
(8)

At a critical point, y=-1 would imply z=0 and p_y=-2, a contradiction. Otherwise p_x=0 implies h=0, after which p_z=0 gives z=x=0 and p_y=0 gives y=0.

See also

Critical Point, Global Maximum, Local Maximum, Partial Derivative

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References

Aduén, H. "The Local-to-Global Minimum Problem of Calvert and Vamanamurthy: A Quartic Counterexample in Three Variables and the Cubic Case in All Dimensions." 1 Nov 2026. https://arxiv.org/abs/2611.02068v1.Anton, H. Calculus: A New Horizon, 6th ed. New York: Wiley, 1999.Apostol, T. M.; Mugler, D. H.; Scott, D. R.; Sterrett, A. Jr.; and Watkins, A. E. A Century of Calculus, Part II: 1969-1991. Washington, DC: Math. Assoc. Amer., 1992.Ash, A. M. and Sexton, H. "A Surface with One Local Minimum." Math. Mag. 58, 147-149, 1985.Calvert, B. and Vamanamurthy, M. K. "Local and Global Extrema for Functions of Several Variables." J. Austral. Math. Soc. 29, 362-368, 1980.Davies, R. "Solution to Problem 1235." Math. Mag. 61, 59, 1988.Rosenholtz, I. and Smylie, L. "The Only Critical Point in Town Test." Math. Mag. 58, 149-150, 1985.Wagon, S. "Failure of the Only-Critical-Point-in-Town Test." §3.4 in Mathematica in Action. New York: W. H. Freeman, pp. 87-91 and 228, 1991.

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Only Critical Point in Town Test

Cite this as:

Weisstein, Eric W. "Only Critical Point in Town Test." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OnlyCriticalPointinTownTest.html

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