Find the Extreme Values of the Function Subject to the Given

Multiple Choice

Find the extreme values of the function subject to the given constraint.
- $f ( x , y ) = x y , \quad 9 x ^ { 2 } + 4 y ^ { 2 } = 36$

A) Maximum: 3 at $\left( \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$
B) Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$ and $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$
C) Maximum: 3 at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( - \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right)$
D) Maximum: 3 at $\left\{ \sqrt { 2 } , \frac { 3 } { 2 } \sqrt { 2 } \right) ;$ minimum: $- 3$ at $\left( \sqrt { 2 } , - \frac { 3 } { 2 } \sqrt { 2 } \right)$