243.14.7.5 Find all local extreme values

• MHB
• karush
In summary, the conversation discusses finding the local extreme values of a given function and identifying them as a local maximum, local minimum, or saddle point. It provides an example of a function and calculates the critical point, as well as the second partials test to determine whether it is an extremum or saddle point. The final answer is a saddle point at (-3/2, 3/2) with a value of 9/2.
karush
Gold Member
MHB
$\tiny{243.14.7.5}$

$\textsf{Find all local extreme values of the given function and identify each}$
$\textsf{as a local maximum,local minimum,or saddlepoint}$
$\textit{$f$has a} \textbf{ saddle point } \textit{at$(a,b)$if}$
\begin{align*} \displaystyle
f_{xx}f_{yy}-fxy^2&\ge 0 \text{ at } (a,b)
\end{align*}
\begin{align*} \displaystyle
f_5(x,y)&=2xy-3x+3y\\
f_x&=2y-3\\
f_y&=2x+3\\
f_x(3/2)&=2y-3=0\\
f_y(-3/2)&=2x+3=0\\
f\left[-\frac{3}{2},\frac{3}{2} \right]
&=2\left[-\frac{3}{2} \cdot \frac{3}{2} \right]
-3\left[-\frac{3}{2} \right]
+3\left[\frac{3}{2} \right]=\frac{9}{2}\\
f_{xx}&=2\\
f_{yy}&=2\\
f_{xx}f_{yy}&=2 \cdot 2-[2 \cdot 2]^2\\
&=4-16=-12 < 0
\end{align*}
\begin{align*} \displaystyle
f\left[-\frac{3}{2},\frac{3}{2} \right]&=\color{red}{\frac{9}{2}}
\end{align*}first time
so suggestions?

We are given:

$$\displaystyle f(x,y)=2xy-3x+3y$$

Identify the critical point(s):

$$\displaystyle f_x(x,y)=2y-3=0\implies y=\frac{3}{2}$$

$$\displaystyle f_y(x,y)=2x+3=0\implies x=-\frac{3}{2}$$

And so our critical point is:

$$\displaystyle (a,b)=\left(-\frac{3}{2},\frac{3}{2}\right)$$

Now, according to the second partials test, we let:

$$\displaystyle D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-\left[f_{xy}(x,y)\right]^2$$

And we find:

$$\displaystyle D(a,b)=0\cdot0-(2)^2=-4<0$$

Hence, we conclude that $f(a,b)$ is not an extremum (it is a saddle point).

1. What is the purpose of finding all local extreme values?

The purpose of finding all local extreme values is to identify the highest and lowest points of a function or data set. This can provide important information about the behavior and characteristics of the function or data.

2. How do you calculate local extreme values?

To calculate local extreme values, you must first take the derivative of the function or data set. Then, set the derivative equal to zero and solve for the variable. The resulting values are the critical points, which can then be plugged back into the original function to determine if they are local maximum or minimum values.

3. What is the difference between a local maximum and a local minimum?

A local maximum is the highest point in a small region of a function or data set, while a local minimum is the lowest point in a small region. In other words, a local maximum is a peak and a local minimum is a valley.

4. Can there be more than one local maximum or minimum?

Yes, there can be multiple local maximum and minimum points in a function or data set. This occurs when the function has multiple peaks or valleys in a given region.

5. How can finding local extreme values be useful in real-life applications?

Finding local extreme values can be useful in many real-life applications, such as analyzing stock market trends, determining the most efficient route for transportation, and optimizing manufacturing processes. It can also help in identifying key data points and making informed decisions based on the behavior of the function or data set.

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