Question

Maximize Q = xy2, where x and y are positive numbers, such that x + y2 = 4.

Answer 1

does y2 means y squared?

Answer 2

yes

Answer 3

If so then the solution of the problem is:
You want to maximize Q, that means you need to find the derivative and critical points of Q. The problem here is that Q is in terms of x and y. You should make it in term of only one of them. So we are going to use the second equation to do so.
\[x+y^2=4 \implies y^2=4-x\]
substitute for the value of y^2 into Q, you get:
\[Q=xy^2=x(4-x)=4x-x^2\]

Answer 4

Do you know how to maximize 4x-x^2?

Answer 5

I plug test points into 4x-x^2?

Answer 6

the critical number is 4?

Answer 7

^^ yeah exactly.

Answer 8

so the max is 3?

Answer 9

you solved for x when the derivative is zero, and you got tgat x=4.
this is your only critical point, so it's either maximum or minimum. Which one do you think it is?

Answer 10

oooh. i dont know

Answer 11

hey wait!!

Answer 12

I made a mistake :(

Answer 13

lol oh no

Answer 14

the critical point is not x=4

Answer 15

I didn't make it, you did and I didn't see it :P

Answer 16

Don't worry.. It's so easy. Do you know how to find the derivative of 4x-x^2?

Answer 17

4- 2x

Answer 18

Right!! Now solve for x when 4-2x=0.

Answer 19

2!

Answer 20

haha so hard

Answer 21

LOL

Answer 22

Now 2 is your critical point, and at x=2 Q is a maximum value.

Answer 23

So now just plug x=2 in Q=4x-x^2, and you're done!!

Answer 24

yay thanks

Answer 25

What is the maximum value?

Answer 26

4

Answer 27

:) I hope you can do any similar problem next time!

Answer 28

thanks for your help!!

Answer 29

no problem.