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

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In summary, we are finding the values of $t$ corresponding to the extrema of $f(x,y,z)=\sin(x^2+y^2)\cos(z)$ subject to the constraints $x^2+y^2=4t, 0\le t\le\pi$, and $z=\frac{\pi}{4}$. The extremum is classified as a minimum when $t=\frac{\pi}{8}$. This is found by setting the derivative equal to 0 and solving for $t$.
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karush
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$\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*}
 
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  • #2
karush said:
$\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...
 

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

1. What is the purpose of finding the value(s) of t corresponding to the extrema?

The purpose of finding the value(s) of t corresponding to the extrema is to identify the critical points of a function, which are points where the function reaches its highest or lowest values. This information can be useful in understanding the behavior of the function and making predictions about its graph.

2. How do you find the value(s) of t corresponding to the extrema?

To find the value(s) of t corresponding to the extrema, you can use the first and second derivatives of the function. The first derivative tells you where the function is increasing or decreasing, and the second derivative tells you where the function is concave up or concave down. The critical points occur at the points where the first derivative is equal to zero or undefined, and the second derivative changes sign.

3. Can there be multiple value(s) of t corresponding to the extrema?

Yes, there can be multiple value(s) of t corresponding to the extrema. These can occur at points where the first derivative is equal to zero or undefined, and the second derivative changes sign. In some cases, the function may have multiple extrema, such as a local maximum and a local minimum.

4. How do you determine if the value(s) of t correspond to a maximum or minimum?

To determine if the value(s) of t correspond to a maximum or minimum, you can use the second derivative test. If the second derivative is positive, the function has a minimum at that point. If the second derivative is negative, the function has a maximum at that point. If the second derivative is zero, the test is inconclusive and you may need to use other methods to determine the type of extrema.

5. Why is it important to find the value(s) of t corresponding to the extrema?

It is important to find the value(s) of t corresponding to the extrema because they can provide valuable information about the behavior of a function. They can help us identify key points on the function's graph, such as turning points or points of inflection. They can also be used to solve optimization problems and make predictions about the behavior of a system.

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