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The only critical point in town test states that a univariate real function 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 . Such exceptions include

(Rosenholtz and Smylie 1985, Wagon 1991).

A related example concerning mixed partial derivatives is

This function has a saddle point at the origin, since its values have opposite signs along and . Direct differentiation gives the unequal mixed partial derivatives and . Both mixed partial derivatives are discontinuous at the origin.

For polynomials in three variables, the quartic

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

as , so this local minimum is not global. To check uniqueness, put . The equations for the gradient are

At a critical point, would imply and , a contradiction. Otherwise implies , after which gives and gives .

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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