Subtopic: Maxima and minima of function with several independent variables
Question Statement:
For a function $f(x, y)$, let $(a, b)$ be a stationary point ($f_x = f_y = 0$). Let $r = f_{xx}, s = f_{xy}, t = f_{yy}$. A saddle point occurs at $(a, b)$ when:
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
Second derivative test: If $rt - s^2 < 0$, the discriminant is negative and the stationary point is a saddle point.