Find the absolute maximum and minimum values of f on the set D. f(x, y) = x4 + y4 − 4xy + 2, D = {(x, y) | 0 ≤ x ≤ 3, 0 ≤ y ≤ 2}

Answer:

Absolute maximum at (3,0)

Absolute minimum at (1,1)

Step-by-step explanation:

Given - f(x, y) = x⁴ + y⁴ − 4xy + 2, D = {(x, y) | 0 ≤ x ≤ 3, 0 ≤ y ≤ 2}

To find - Find the absolute maximum and minimum values of f on the set D.

Proof -

Given that, f(x, y) = x⁴ + y⁴ − 4xy + 2

Now,

Differentiate f(x,y) with respect to x and y , we get

[tex]f_{x}(x,y)[/tex] = 4x³ - 4y

[tex]f_{y}(x,y)[/tex] = 4y³ - 4x

Now,

For the critical point, put [tex]f_{x}(x,y)[/tex] , [tex]f_{y}(x,y)[/tex]  = 0

we get

4x³ - 4x = 0

⇒x = 0, -1, 1

but given the range of x is 0 ≤x ≤ 3

⇒x = 0, 1

⇒y = 0, 1

∴ we get

2 critical points (x,y) = (0,0) , (1,1)

Now,

Put the values of x and y in f(x,y), we get

f(0,0) = 0 + 0 - 0 + 2 = 2

f(1,1) = 1 + 1 - 4 + 2 = 0

Now,

for the max and min values, finding the critical value is not enought, we have to find the values in the domain also.

Given domain is 0 ≤ x ≤ 3, 0 ≤ y ≤ 2

So, we have to check the values at the rectangle that have the coordinates

(0,0), (0,2), (3,2), (3,0)

Now,

f(0,0) = 0 + 0 - 0 + 2 = 2

f(0,2) = 0 + 16 - 0 + 2 = 18

f(3,2) = 81 + 16 - 24 + 2 = 75

f(3,0) = 81 + 0 - 0 + 2 = 83

∴ we get

Absolute maximum at (3,0)

Absolute minimum at (1,1)

1. Tamekia and Marsha mow lawns during the summer to earn money. Tamekia
determined that she can earn between $6.00 and $6.25 per hour. Marsha
estimates that she earns between $7.50 and $8.00 per hour. About how much
more money will Marsha earn than Tamekia if they each work 22 hours? *
A. $65.01 to $85.00
B. $45.01 to $65.00
C. $25.01 to $45.00
D. $5.00 to $25.00

The sales rate is 8%. If Janelle buys a gymnastics mat priced at $87 , how much tax will she pay?

Hey guys, I just got braces and they are really uncomfy so how long did it take yall to get used to them?

PLs help me. I need to find the measure of B

what is the slope of 4x-3y=18

WILL MARK BRAINLY FOR WHO EVER GETS THIS RIGHT

Pleaseee helppp me on this

please help i have been stuck on this for 3 hours

-7x + 2y = -5
y = -7x + 8
explain how please

Problem Suppose an archaeologist finds a camel tooth that contains 42 % of original amount of C-14. Find age of the tooth. N = Noekt No = initial mass of C-14 (at time t= 0) N = amount of C-14 at time t k= 0.0001 t = time, in years Since, our equation can be written like this: 0.42 No = Noe Noe -0.00010​

Please help.. You can answer one if you want but whoever answers both get brainliest

The area of a rectangle with a
length of 10 cm and a width of 3
cm.

Anyone who know what the answer please please?!

I REALLLY NEEED THISSSS ANSWERRRR PLZZZZ I'M GOING TO FAILLL I ONLY NEEEED ONE MOREEEE.
Give an example of other data that can
be represented using a histogram.
________________________________________
________________________________________

giving out brainliest answer !!!! help me out asap please !!!!

Rachel has a drawerful of pop, rock, metal, and country CDs. The table gives the probability of randomly picking a CD of each type from the
drawer.
Music Style Probability of Picking a CD
pop
0.2
rock
0.1
metal
0.3
country
0.4
Rachel picks a CD 150 times, replacing each pick before the next draw. Arrange the events in order from the lowest predicted frequency to the
highest predicted frequency.
picking a pop
music CD
picking a rock
music CD
picking a metal
music CD
picking a country
music CD

How many 4 digit numbers are multiples of 8?

Which type of graph(s) should be used for the following data? Choose all that apply. (Check photo)

PLEASE HELP !! ILL GIVE BRAINLIEST !! 100 POINTS

What is the length of missing side Y Options are on photo

what is -3/4 +7/2 in simplest form

A box of donuts cost $9. You want to send donuts to the local nursing home. Set up an equation to find how many boxes you can send if you have $72.

I need help if you get right I will mark you

At a school carnival, 51 out of 68 tickets sold were early-admission tickets. What percentage of the tickets were early-admission tickets?

set up a proportion and use it to solve for x​

Jill wants to paint the front and top surfaces of each step leading to her porch. There are 6 steps like the one shown. How many total square inches of the steps will she paint?

A: 1,296

B: 1,776

C: 3,552

D: 5,760​

What is the volume if
the base is 48 square inches and height is 8 inches

Find the distance between the points (4.5,-1.5), (4.5,7.25)
The distance is ____.

izllllllllllllllllllll

(please answer if you know) On a​ map, 1 inch equals 5.2 miles. Two houses are 3.5 inches apart on the map. What is the actual distance between the​ houses? Use pencil and paper. Show how you can represent the scale with two different ratios. What ratio is more helpful for solving the​ problem? Explain.