Clairaut's theorem (Schwarz's theorem) guarantees the equality of mixed second-order partial derivatives $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ at a point $(a, b)$ provided that:
Options:
(A) Both mixed partial derivatives are continuous in an open disk containing $(a, b)$
(B) The function $f$ is bounded everywhere
(C) $f(x, y)$ is a polynomial in $x$ and $y$
(D) The gradient of $f$ vanishes at $(a, b)$
Answer Key:
A
Step-by-Step Solution & Explanation
Clairaut's theorem on the equality of mixed partials states that if $f(x, y)$ and its partial derivatives $f_x, f_y, f_{xy}$, and $f_{yx}$ exist and the mixed partials $f_{xy}$ and $f_{yx}$ are continuous throughout an open region containing $(a, b)$, then $f_{xy}(a, b) = f_{yx}(a, b)$.