find critical points and relative extrema, given an open region.

sibuzwaW
2020-10-27
Answered

Relative extrema of multivariables:

$f(x,y)=\frac{xy}{7}$

find critical points and relative extrema, given an open region.

find critical points and relative extrema, given an open region.

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Clara Reese

Answered 2020-10-28
Author has **120** answers

We have,

To find the critical point:

Find the partial derivative and equate to zero:

Now,

Differntiate f(x,y) with respect to x and equate to zero

Differntiate f(x,y) with respect to y and equate to zero

Hence, the critical point is (0,0)

Now we have to find the relative extrema.

Use:

Find D(0,0):

Now

Therefore,

Hence, by the

f(x,y) neither max. nor min at (0,0) which means (0,0) is saddle point.

asked 2021-09-08

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.

$P\left(x\right)=-12{x}^{2}+2136x-41000$

To maximize the monthly rental profit , how many units should be rented out?

What is the maximum monthly profit realizable?

To maximize the monthly rental profit , how many units should be rented out?

What is the maximum monthly profit realizable?

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Use polar coordinates to find the limit. [Hint: Let $x=r\mathrm{cos}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}y=r\mathrm{sin}$ , and note that (x, y) (0, 0) implies r 0.]
$\underset{(x,y)\to (0,0)}{lim}\frac{{x}^{2}-{y}^{2}}{\sqrt{{x}^{2}+{y}^{2}}}$

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Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes’ Theorem becomes the circulation form of Green’s Theorem.

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I have a very simple linear problem:

$\begin{array}{rl}\underset{x}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}+{a}_{2}{x}_{2}=b\end{array}$

Suppose I want to write this problem equivalently as in Find the equivalent linear program. Unlike the problem in the link, I have equality. Can I write it equivalently as:

$\begin{array}{rl}\underset{x,\alpha ,\beta}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}=\alpha b,\text{}{a}_{2}{x}_{2}=\beta b,\text{}\alpha +\beta =1.\end{array}$

The converse is intuitive: Given $\{x,\alpha ,\beta \}$ feasible for the second problem, adding the first and second constraints gives the constraint of the first problem. But the forward part is not clear, especially because I have never seen an equality constraint written like this. Any help would be highly appreciated.

$\begin{array}{rl}\underset{x}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}+{a}_{2}{x}_{2}=b\end{array}$

Suppose I want to write this problem equivalently as in Find the equivalent linear program. Unlike the problem in the link, I have equality. Can I write it equivalently as:

$\begin{array}{rl}\underset{x,\alpha ,\beta}{min}& \text{}{x}^{2}\\ \text{s.t.}& \text{}{a}_{1}{x}_{1}=\alpha b,\text{}{a}_{2}{x}_{2}=\beta b,\text{}\alpha +\beta =1.\end{array}$

The converse is intuitive: Given $\{x,\alpha ,\beta \}$ feasible for the second problem, adding the first and second constraints gives the constraint of the first problem. But the forward part is not clear, especially because I have never seen an equality constraint written like this. Any help would be highly appreciated.

asked 2021-03-02

COnsider the multivariable function $g(x,y)={x}^{2}-3{y}^{4}{x}^{2}+\mathrm{sin}\left(xy\right)$ . Find the following partial derivatives: ${g}_{x}.{g}_{y},{g}_{xy},g(\times ),g\left(yy\right)$ .

asked 2020-12-13

Use Greens

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use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F and the curve C.

C: The square bounded by