Calculus

Clairaut's theorem (Schwarz's theorem) guarantees equality of mixed second partial derivatives $\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ provided:

Options:

• (A) The second partial derivatives are continuous in a neighborhood of the point
• (B) $f(x, y)$ is a polynomial of degree 1
• (C) $\nabla f = 0$ everywhere
• (D) $x = y$ identically