Mathematics, 22.04.2020 22:18, AdoNice

We want to find the maximum and minimum values of f(x, y)=12x2+13y2 on the disk D: x2+y2≤1. What is the critical point in D? (x, y)=( , ) Now focus on the boundary of D, and solve for y2. Restricting f(x, y) to this boundary, we can express f(x, y) as a function of a single variable x. What is this function and its closed interval domain? f(x, y)=f(x)= where ≤x≤ What are the absolute maximum and minimum values of the function along the BOUNDARY of D? maximum value: minimum value: What are the absolute maximum and minimum values of f(x, y) over all of D? maximum value: minimum value:

Answers: 2

Show answers

Answers

answered: youngsiron

Over the boundary: maximum:13, minimum:12

Over D: maximum:13, minimum:0

Step-by-step explanation:

We are given that f(x,y) = 12x^2+13y^2 and D is the disc of radius one. Namely, $x^2+y^2\leq 1$ .

First, we want to find a critical point of the function f. To do so, we want to find the values(x,y) such that

$\nabla f (x,y) =0$ .

Recall that $\nabla f (x,y) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})$ .

So, let us calculate $\nabla f (x,y)$ (the detailed calculation of the derivatives is beyond the scope of the answer.

$\frac{\partial f}{\partial x} = 24x$

$\frac{\partial f}{\partial y} = 26y$

When equalling it to 0, we get that the critical point is (0,0), which is in our region D. Note that the function f is the sum of the square of two real numbers multiplied by some constants. Hence, the function f fulfills that $f(x,y)\geq 0$ . Note that f(0,0)=0, so without further analysis we know that the point (0,0) is a minimum of f over D.

If we restrict to the boundary, we have the following equation x^2+y^2=1 . Main idea is to replace the value of one of the variables n the function f, so it becomes a function of a single variable. Then, we can find the critical values by using differential calculus:

Case 1:

Let us replace y. So, we have that y^2=1-x^2 . So, f(x,y) = 12x^2+13(1-x^2) = -x^2+13 .

So, we will find the derivative with respect to x and find the critical values. That is

$\frac{df}{dx} = -2x=0$

Which implies that x =0. Then, $y =\pm 1$ . So we have the following critical points (0,1), (0,-1). Notice that for both points, the value of f is f(0,1) = f(0,-1) = 13. If we calculate the second derivative, we have that at x=0

$\frac{d^2f}{dx^2} = -2$ . By the second derivative criteria, we know that this points are local maximums of the function f.

Case 2:

Let us replace x. So, we have that x^2=1-y^2 . So, f(x,y) = 12(1-y^2)+13y^2 = y^2+12 .

So, we will find the derivative with respect to y and find the critical values. That is

$\frac{df}{dx} = 2y=0$

Which implies that y =0. Then, $x =\pm 1$ . So we have the following critical points (-1,0), (1,0). Notice that for both points, the value of f is f(1,0) = f(-1,0) = 12. If we calculate the second derivative, we have that at y=0

$\frac{d^2f}{dx^2} = 20$ . By the second derivative criteria, we know that this points are local minimums of the function f.

So, over the boundary D, the maximum value of f is 13 and the minimum value is 12. Over all D, the maximum value of f is 13 and the minimum value is 0.

answered: Guest

attachment?

step-by-step explanation:

answered: Guest

the office resources are the human, physical and financial resources which are utilized for performing all the activities in order to achieve the organizational objectives.

happy to

pls mark as brainliest

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 14:00, jannaleigh

Jackie bought a bag of oranges. she used 3/8 of the oranges to make orange juice. she used 1/3 of the oranges to make a fruit salad. she put 1/8 of the oranges in a bowl on the table. she put 2/12 of the oranges in the refrigerator. a) jackie left more oranges on the table than she put in the refrigerator. b) jackie used fewer oranges to make orange juice than she left on the table. c) jackie put fewer oranges in the refrigerator than she used to make the fruit salad. eliminate d) jackie used more oranges to make a fruit salad than she used to make orange juice.

Answers: 1

continue

Mathematics, 21.06.2019 17:00, samiam61

The equation of a circle is (x + 4)2 + (y + 6)2 = 16. determine the length of the radius. 10 4 16 8 my flower basket delivers flowers within a 20 mile radius of their store. if this area is represented graphically, with my flower basket located at (5, 2), what is the equation that represents the delivery area? (x - 2)2 + (y - 5)2 = 20 (x - 5)2 + (y - 2)2 = 20 (x + 5)2 + (y + 2)2 = 400 (x - 5)2 + (y - 2)2 = 400

Answers: 1

continue

Mathematics, 21.06.2019 17:30, jacamron

Can someone me and do the problem plz so i can understand it more better

Answers: 2

continue

Mathematics, 21.06.2019 17:30, neverfnmind

James adds two numbers a + b. sally adds the same two numbers but reverse the order b + a what property of addition assures us that james and sally will get the same sum

Answers: 2

continue

Do you know the correct answer?

We want to find the maximum and minimum values of f(x, y)=12x2+13y2 on the disk D: x2+y2≤1. What is...