Subtopic: Maxima and minima of function with several independent variables
Question Statement:
For a function $f(x, y)$ with stationary point $(a, b)$, let $r = f_{xx}, s = f_{xy}, t = f_{yy}$. A saddle point occurs if:
Options:
(A) $r t - s^2 > 0$ and $r > 0$
(B) $r t - s^2 > 0$ and $r < 0$
(C) $r t - s^2 < 0$
(D) $r t - s^2 = 0$
Answer Key:
C
Step-by-Step Solution & Explanation
The discriminant is $D = r t - s^2$:
- If $D > 0$ and $r > 0$: Local minimum.
- If $D > 0$ and $r < 0$: Local maximum.
- If $D < 0$: Saddle point.
- If $D = 0$: Test is inconclusive.