Provide an Appropriate Response $t$ Corresponding to the Extrema Of $f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )$

Multiple Choice

Provide an appropriate response.
-Find the value(s) of $t$ corresponding to the extrema of $f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )$ subject to the constraints $x ^ { 2 } +$ $y ^ { 2 } = 6 t , 0 \leq t \leq \pi$ , and $z = \frac { \pi } { 6 }$ . Classify each extremum as a minimum or maximum. (Hint: $w = f ( x , y , z )$ is a differentiable function of t.

A) $t = \frac { \pi } { 12 }$ , minimum; $t = 0$ , maximum
B) $t = \frac { \pi } { 12 }$ , minimum
C) $\mathrm { t } = - \frac { \pi } { 12 }$ , maximum
D) $\mathrm { t } = \frac { \pi } { 12 }$ , minimum; $\mathrm { t } = - \frac { \pi } { 12 }$ , maximum