Subtopic: Maxima and minima of function with several independent variables
Question Statement:
For a function $f(x, y)$ with stationary point $(a, b)$, let $r = f_{xx}(a, b)$, $s = f_{xy}(a, b)$, and $t = f_{yy}(a, b)$. The point $(a, b)$ is a SADDLE POINT if:
Options:
(A) $r t - s^2 < 0$
(B) $r t - s^2 > 0$ and $r > 0$
(C) $r t - s^2 > 0$ and $r < 0$
(D) $r t - s^2 = 0$
Answer Key:
A
Step-by-Step Solution & Explanation
By the second derivative test for functions of two variables:
- If $rt - s^2 > 0$ and $r > 0$, $(a,b)$ is a local minimum.
- If $rt - s^2 > 0$ and $r < 0$, $(a,b)$ is a local maximum.
- If $rt - s^2 < 0$, $(a,b)$ is a saddle point.