Subtopic: Maxima and minima of function with several independent variables
Question Statement:
Let $(a, b)$ be a stationary point of a twice continuously differentiable function $f(x, y)$ with $r = f_{xx}(a,b)$, $s = f_{xy}(a,b)$, and $t = f_{yy}(a,b)$. Which of the following statements is/are TRUE?
Options:
(A) If $rt - s^2 > 0$ and $r > 0$, $f(a, b)$ is a local minimum
(B) If $rt - s^2 > 0$ and $r < 0$, $f(a, b)$ is a local maximum
(C) If $rt - s^2 < 0$, $(a, b)$ is a saddle point
(D) If $rt - s^2 = 0$, the second derivative test is inconclusive
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By the second derivative test for functions of two variables with Hessian determinant $\Delta = rt - s^2$:
1. $\Delta > 0$ and $r > 0 \implies$ local minimum (A is true).
2. $\Delta > 0$ and $r < 0 \implies$ local maximum (B is true).
3. $\Delta < 0 \implies$ saddle point (C is true).
4. $\Delta = 0 \implies$ inconclusive test, requiring higher order terms or alternative analysis (D is true).