Subtopic: Maxima and minima of function with several independent variables
Question Statement:
Let $(x_0, y_0)$ be a stationary point of a twice continuously differentiable function $f(x, y)$, where $\nabla f(x_0, y_0) = 0$. Let $r = f_{xx}$, $s = f_{xy}$, $t = f_{yy}$ at $(x_0, y_0)$, and $D = rt - s^2$. Which of the following statements is/are CORRECT?
Options:
(A) If $D > 0$ and $r > 0$, then $f$ has a local minimum at $(x_0, y_0)$
(B) If $D > 0$ and $r < 0$, then $f$ has a local maximum at $(x_0, y_0)$
(C) If $D < 0$, then $f$ has a saddle point at $(x_0, y_0)$
(D) If $D = 0$, the second derivative test is inconclusive
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
The discriminant $D = rt - s^2 = \det(\mathcal{H})$ is the determinant of the Hessian matrix $\mathcal{H}$:
1. If $D > 0$, both eigenvalues of $\mathcal{H}$ share the same sign as $r$. If $r > 0$, $\mathcal{H}$ is positive definite (local minimum, Option A is true). If $r < 0$, $\mathcal{H}$ is negative definite (local maximum, Option B is true).
2. If $D < 0$, the eigenvalues have opposite signs, making $\mathcal{H}$ indefinite (saddle point, Option C is true).
3. If $D = 0$, at least one eigenvalue is zero, making $\mathcal{H}$ semi-definite or degenerate, so higher-order terms must be examined (inconclusive, Option D is true).