Subtopic: Maxima and minima of function with several independent variables
Question Statement:
For a two-variable scalar function $f(x, y)$ with continuous second-order partial derivatives, let $(a, b)$ be a stationary point ($f_x = 0, f_y = 0$). Define $D = f_{xx} f_{yy} - (f_{xy})^2$. Which of the following statements is/are CORRECT?
Options:
(A) If $D > 0$ and $f_{xx} > 0$, then $f(x, y)$ has a local minimum at $(a, b)$
(B) If $D > 0$ and $f_{xx} < 0$, then $f(x, y)$ has a local maximum at $(a, b)$
(C) If $D < 0$, then $(a, b)$ is a saddle point of $f(x, y)$
(D) If $D = 0$, the second derivative test is inconclusive
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
Multivariable Second Derivative Test:
- $D > 0, f_{xx} > 0 \implies$ Local Minimum (A is true).
- $D > 0, f_{xx} < 0 \implies$ Local Maximum (B is true).
- $D < 0 \implies$ Saddle point (C is true).
- $D = 0 \implies$ Test inconclusive (D is true).
All four options are correct.