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

Question

Quizplus · Accepted Answer

Given $f(x, y, z) = \sin(x^2 + y^2)\cos(z)$ with constraints $x^2 + y^2 = 6t$ and $z = \frac{\pi}{6}$, we substitute the constraints into $f$ to get $f(t) = \sin(6t)\cos(\frac{\pi}{6})$. The derivative of $f(t)$ with respect to $t$ will give us the critical points. Since $\cos(\frac{\pi}{6})$ is positive, the extrema of $f(t)$ depend on the sine component. The first derivative of $\sin(6t)$ will have critical points where its derivative is zero, leading to $t = \frac{\pi}{12}$ as a critical point within the given interval $0 \leq t \leq \pi$. The second derivative test or analyzing the behavior of the function around this point will show that it is a minimum because the sine function increases from this point within the given interval. There is no maximum within the given constraints and interval for $t$.

Provide an Appropriate Response ttt Corresponding to the Extrema Of f(x,y,z)=sin⁡(x2+y2)cos⁡(z)f ( x , y , z ) = \sin \left( x ^ { 2 } + y ^ { 2 } \right) \cos ( z )f(x,y,z)=sin(x2+y2)cos(z)

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