Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function $f ( x , y )$ at the critical point $\left( x _ { 0 } , y _ { 0 } \right)$ . Given: $\begin{array} { l }
f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = - 1 \
f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 8 \
f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = 5
\end{array}$
A) $\text { relative minimum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
B) $\text { saddle point }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
C) $\text { relative maximum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
D) $\text { insufficient information to determine the nature of the function }$ at $\left( x _ { 0 } , y _ { 0 } \right)$

Question

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function $f ( x , y )$ at the critical point $\left( x _ { 0 } , y _ { 0 } \right)$ . Given: $\begin{array} { l } 
f _ { x x } \left( x _ { 0 } , y _ { 0 } \right) = - 1 \
f _ { y y } \left( x _ { 0 } , y _ { 0 } \right) = - 8 \
f _ { x y } \left( x _ { 0 } , y _ { 0 } \right) = 5
\end{array}$
A) $\text { relative minimum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
B) $\text { saddle point }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
C) $\text { relative maximum }$ at $\left( x _ { 0 } , y _ { 0 } \right)$
D) $\text { insufficient information to determine the nature of the function }$ at $\left( x _ { 0 } , y _ { 0 } \right)$

Quizplus · Accepted Answer

To determine the nature of a critical point for a function of two variables, we use the second partial derivative test, which involves calculating the determinant of the Hessian matrix at the critical point: $D = f_{xx}f_{yy} - (f_{xy})^2$. Substituting the given values, we get $D = (-1)(-8) - (5)^2 = 8 - 25 = -17$. Since $D < 0$, the critical point is a saddle point.

Determine Whether There Is a Relative Maximum, a Relative Minimum f(x,y)f ( x , y )f(x,y)

Determine Whether There Is a Relative Maximum, a Relative Minimum $f ( x , y )$