Suppose (1, 1) is a critical point of a function f with continuous second derivatives. In the case of $f _ { x } ( 1,1 ) = 8$ , $f _ { x y } ( 1,1 ) = 8$ , $f _ { y y } ( 1,1 ) = 10$ what can you say about f ?
A) f has a local maximum at (1,1)
B) f has a saddle point at (1,1)
C) f has a local minimum at (1,1)

Question

Quizplus · Accepted Answer

Since $f_{xx}(1,1)>0$, we know that $(1,1)$ is a critical point and that $f$ has a local extremum at $(1,1)$. Since $f_{xx}(1,1)f_{yy}(1,1)-[f_{xy}(1,1)]^2=(8)(10)-[8]^2<0$, we know that the extremum is not a saddle point. Therefore, the only remaining possibility is that $f$ has a local minimum at $(1,1)$.

Suppose (1, 1) Is a Critical Point of a Function fx(1,1)=8f _ { x } ( 1,1 ) = 8fx​(1,1)=8

Suppose (1, 1) Is a Critical Point of a Function $f _ { x } ( 1,1 ) = 8$