The Function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$

The function $f ( x , y ) = 4 - x ^ { 2 } - y ^ { 2 } - x y - x$ has one critical point. Determine its location and type.

A) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , saddle point
B) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , minimum point
C) $\left( - \frac { 2 } { 3 } , \frac { 1 } { 3 } \right)$ , maximum point
D) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , saddle point
E) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , minimum point
F) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } \right)$ , maximum point
G) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , saddle point
H) $\left( \frac { 1 } { 3 } , - \frac { 2 } { 3 } \right)$ , minimum point

Correct Answer:

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