Question

If the function f(x, y) has mixed second order partial derivatives $f_{xy$ and $f_{yx}$ that are continuous in an open disk containing ($x_0, y_0$), then :}

• A. $f_{yx}(x_0, y_0) \neq f_{xy}(x_0, y_0)$
• B. $f_{yx}(x_0, y_0) = f_{xy}(x_0, y_0)$
• C. $f_{yx}(x_0, y_0) > f_{xy}(x_0, y_0)$
• D. $f_{yx}(x_0, y_0) < f_{xy}(x_0, y_0)$

Hint:

For almost all smooth, continuous functions encountered in physics and engineering, mixed partial derivatives are equal ($f_{xy} = f_{yx}$).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question concerns Clairaut's Theorem (also known as Schwarz's Theorem) on the equality of mixed partial derivatives.
Detailed Explanation:
Let us review the theorem's statement:
- Let $f$ be a function of two variables $x$ and $y$.
- The mixed second-order partial derivatives are $f_{xy} = \frac{\partial^2 f}{\partial y \partial x}$ and $f_{yx} = \frac{\partial^2 f}{\partial x \partial y}$.
- Clairaut's Theorem states that if $f$ is defined on an open disk containing a point $(x_0, y_0)$, and the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are both continuous on this disk, then: \[ f_{yx}(x_0, y_0) = f_{xy}(x_0, y_0) \] - The order of differentiation does not matter under the condition of continuity.

Step 2: Final Answer:
The correct relationship is $f_{yx}(x_0, y_0) = f_{xy}(x_0, y_0)$, which corresponds to Option (B).