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

question 12

Multiple Choice

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 \(\left( x _ { 0 } , y _ { 0 } \right)\)

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