Given $f(x,y)=2x^4-xy^2+2y^2,0\le x\le 4, 0\le y\le2$. Find absolute extrema of $f(x,y)$.

You are watching: Find the absolute maxima and minima of the function on the given domain.

I have uncovered $\partial f/\partial x=8x^3-y^2, \partial f/\partial y=-2xy+4y$ and after fixing the equation through letting $\partial f/\partial x=0$ and $\partial f/\partial y=0$, the an essential point room $(0,0), (2,-8) and also (2,-8)$. I"m lost just how to discover the absolute extrema, isn"t any kind of alternative means to resolve this question? You deserve to follow a recipe because that these questions.

$\quad(1)$: If the preferably or minimum lies ~ above the internal of the domain, climate it need to be a an essential point (that is, its gradient must vanish).

$\quad\quad(1.1)$: To recognize whether a crucial point is a neighborhood maximum or a regional minimum (or saddle), friend may directly compute the values and also compare them, or else employ a greater order test (e.g. Hessian).

$\quad(2)$: If the best or minimum lies on the boundary of the domain, you might use Lagrange multipliers to uncover them.

$\quad\quad(2.1)$: as soon as the boundary of the domain is $1$-dimensional, you might parametrize and also min-max follow me the parametrization; this to reduce the step to a single-variable calculus exercise.

In general, us don"t know a priori even if it is $(1)$ or $(2)$ applies, therefore we need to examine for extrema both in the interior and in the border of the domain.

re-superstructure
mention
follow
answered might 17 "18 in ~ 20:08 FimpellizieriFimpellizieri
$\endgroup$
0
$\begingroup$
Recall that we additionally need come look for the extrema on the boundary, in this case since we have actually not an important points in the interior of the domain the extrema points are on the boundary, then we have to check

$f(0,y)$ and also $f(4,y)$ because that $0\le y\le2$

$f(x,0)$ and also $f(x,2)$ for $0\le x\le4$

share
point out
monitor
answered might 17 "18 in ~ 20:07 useruser
$\endgroup$
0
$\begingroup$
Once you"ve found the extrema, plug them into $f$ and see which one returns the shortest value.

You"ll also need to make sure $f$ isn"t reduced on the border of her domain.

re-superstructure
mention
monitor
answered may 17 "18 at 20:07 NicNic8NicNic8
$\endgroup$
include a comment |
0
$\begingroup$
Since your set $E=\(x,y)\in\princetoneclub.orgbbR^2:0\leq x\leq4,0\leq y\leq2\$ is compact and also $f$ is consistent on $E$ (being a polynomial), we recognize that $f$ suspect a an international minimum and also a global maximum top top $E$.

You can uncover the extrema in the following way.

First friend look for extrema in the interior by calculating the first partial derivatives. Setup both same to zero and solving yields critical points.

This must be sufficient to resolve this exercise.

re-publishing
point out
follow
answered might 17 "18 in ~ 20:12 Václav MordvinovVáclav Mordvinov
$\endgroup$
0
$\begingroup$
You deserve to avoid the constraints making

$$u = 4\left(\frac\sin(\phi)+12\right)\\v = 2\left(\frac\sin(\eta)+12\right)$$

$$\min\max f(\phi,\eta) = 2u^4-uv^2+2v^2 = 2u^4-(u-2)v^2$$

The stationary problems give

$$\nabla f = (f_\phi,f_\eta) = \left\{\beginarrayrcl128 (\sin (\phi )+1)^3 \cos (\phi )-2 (\sin (\eta )+1)^4 \cos (\phi )&=&0\\-8 (\sin (\eta )+1)^3 \cos (\eta ) (\sin (\phi )+1)-4 (\sin (\eta )+1) \cos (\eta )& = & 0\endarray\right.$$

or

$$\left\{\beginarrayrcl\cos(\phi) & = & 0\\\sin(\eta) & = & 0\\64 (\sin (\phi )+1)^3-(\sin (\eta )+1)^4&=&0\\2 (\sin (\eta )+1)^2 (\sin (\phi )+1)+1& = & 0\endarray\right.$$

Calling

$$p = \sin(\phi)+1\\q=\sin(\eta)+1$$

the mechanism last two equations read

$$64p^3-q=0\\2p q^2+1=0$$

and hence the values for $\phi, \eta$ are conveniently obtained

re-publishing
cite
monitor
answered might 17 "18 in ~ 21:20
CesareoCesareo
$\endgroup$
include a comment |

Thanks because that contributing an answer to princetoneclub.orgematics ridge Exchange!

Please be sure to answer the question. Provide details and also share her research!

But avoid

Asking for help, clarification, or responding to various other answers.Making statements based on opinion; ago them increase with references or personal experience.

Use princetoneclub.orgJax to format equations. Princetoneclub.orgJax reference.

See more: System Restore Could Not Find The Offline Boot Volume Windows 10

Draft saved

authorize up utilizing Email and also Password
send

### Post as a guest

name
email Required, however never shown

### Post as a guest

surname
email

Required, yet never shown

## Not the price you're looking for? Browse various other questions tagged multivariable-calculus or ask your own question.

Featured top top Meta
48 votes · comment · stats
13
$X$ compact metric space, $f:X\rightarrow\princetoneclub.orgbbR$ consistent attains max/min
related
3
recognize the absolute max and min of a function bounded by a domain D.
1
uncover absolute maximum and minimum with domain
0
Finding pure Min/Max with offered Domain and also Equation. F(x,y)
1
find absolute maximum and also minimum
1
proving a multi variable duty has an pure maximum and also minimum in one ellipse.
1
uncover the pure minimum and maximum that $f(x,y)=x^2y+2xy+12y^2$ ~ above the ellipse $x^2+2x+16y^2\leq8$
4
Finding pure Minimum and also Absolute preferably of $f(x,y)=xy$
0
find absolute maximum and also minimum through explanation
warm Network inquiries an ext hot concerns

concern feed

princetoneclub.orgematics
firm
ridge Exchange Network
site design / logo design © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.9.10.40187

princetoneclub.orgematics ridge Exchange works best with JavaScript permitted 