Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function $f ( x , y ) \text { at the critical point }$ $\left( x _ { 0 } , y _ { 0 } \right) \text {, if } f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = 16 , f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 2 , f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = - 1$
A) relative maximum
B) relative minimum
C) saddle point
D) insufficient information .

Question

Quizplus · Accepted Answer

Since $f_{xx}(x_0,y_0)>0$ and $f_{yy}(x_0,y_0)<0$, we know that the critical point $(x_0,y_0)$ is a saddle point. The fact that $f_{xy}(x_0,y_0)=-1$ does not give us enough information to determine the nature of the critical point.

Determine Whether There Is a Relative Maximum, a Relative Minimum f(x,y) at the critical point f ( x , y ) \text { at the critical point }f(x,y) at the critical point

Determine Whether There Is a Relative Maximum, a Relative Minimum $f ( x , y ) \text { at the critical point }$