Subtopic: Constrained extrema and Lagrange multipliers
Question Statement:
Let $(a, b)$ be a critical point of a two-variable function $f(x, y)$, and let $r = f_{xx}(a, b)$, $s = f_{xy}(a, b)$, and $t = f_{yy}(a, b)$. Which of the following classifications is/are TRUE?
Options:
(A) If $rt - s^2 > 0$ and $r < 0$, then $f(a, b)$ is a local maximum
(B) If $rt - s^2 > 0$ and $r > 0$, then $f(a, b)$ is a local minimum
(C) If $rt - s^2 < 0$, then $(a, b)$ is a saddle point
(D) If $rt - s^2 = 0$, the test is inconclusive and further investigation is required
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
By the Second Derivative Test for functions of two variables, the Hessian determinant is $\Delta = rt - s^2$:
1. $\Delta > 0$ and $r < 0 \implies$ local maximum. (A is true)
2. $\Delta > 0$ and $r > 0 \implies$ local minimum. (B is true)
3. $\Delta < 0 \implies$ saddle point. (C is true)
4. $\Delta = 0 \implies$ test fails/inconclusive. (D is true)
All choices A, B, C, D are true.