-z.55 Find the value(s) of t corresponding to the extrema

[karush]

$\text{Find the value(s) of $t$ corresponding to the extrema of}$
$$f(x,y,z)=\sin(x^2+y^2)\cos(z)$$
$\text{subject to the constraints} $
$$\text{$x^2+y^2=4t, 0\le t\le\pi$, and $z=\frac{\pi}{4}$}$$
$\text{Classify each extremum as a minimum or maximum.}$
\begin{align*} \displaystyle
f_7(x,y,z)&=\sin(4t)\cos\left(\frac{\pi}{4}\right)\\
&=\frac{\sqrt{2}}{2}\sin(4t)\\
f_7^\prime&=2\sqrt{2}\cos(4t)\\
&\textbf{got lost here}\\
\therefore t&=\color{red}{\frac{\pi}{8} , \textit{min}}
\end{align*}

 

[Prove It]

$\text{Find the value(s) of $t$ corresponding to the extrema of}$
$$f(x,y,z)=\sin(x^2+y^2)\cos(z)$$
$\text{subject to the constraints} $
$$\text{$x^2+y^2=4t, 0\le t\le\pi$, and $z=\frac{\pi}{4}$}$$
$\text{Classify each extremum as a minimum or maximum.}$
\begin{align*} \displaystyle
f_7(x,y,z)&=\sin(4t)\cos\left(\frac{\pi}{4}\right)\\
&=\frac{\sqrt{2}}{2}\sin(4t)\\
f_7^\prime&=2\sqrt{2}\cos(4t)\\
&\textbf{got lost here}\\
\therefore t&=\color{red}{\frac{\pi}{8} , \textit{min}}
\end{align*}

Set the derivative equal to 0 and solve for t...