Question

Calculus

Answer 1

Find two positive numbers x and y such that x+y=60
xy is maximum

Answer 2

@Kainui

Answer 3

Any ideas how to start this or would you like a hint or what? =) This is my favorite type of question.

Answer 4

let me think more

Answer 5

shall we make one equation in y and substitute it in other

Answer 6

Yup. =)

Answer 7

y=60-x
x(60-x)

60x-x^2


first derivative = 60 - 2x

60-2x=0
x=30 -> which is a maxima

both are 30 then

Answer 8

right? or i made a mistake

Answer 9

No, this is perfect. A way you might check yourself is to wonder what if you decreased one? Then the other increases by the same amount, and it's symmetric because they're the same.

But notice that upon lowering one a little bit you go down. http://www.wolframalpha.com/input/?i=29.9*30.1&t=crmtb01

Often times questions with a symmetric question have a symmetric answer. If you think about it, x sort of has to be y because there is no difference between the two. Although this isn't necessarily always true.

Answer 10

oh!
What if it asked minimum x and y value they would still be 30 right

Answer 11

Nope, it would have to be 0 and 60 for one or the other. Since if they have to be positive that means 0*60=0 is the minimum.

Answer 12

but then the derivative test for maxima and minima don't tell us anything
or probably they are just local not global

Answer 13

The only tell you if a point has a slope of 0. This could be a max, min, or an inflection point (like the center of the graph y=x^3).

Generally speaking you should check the points inside and the endpoints. The endpoints are sort of implied by saying "two positive numbers x and y"

Answer 14

i don't get what u mean

Answer 15

i mean i am not getting what do u mean by inside

Answer 16

So this is the graph you came up with and then took the derivative to find the maximum correct?

f(x)=60-x^2

http://www.wolframalpha.com/input/?i=f(x)%3D60x-x%5E2&t=crmtb01

See how on the graph there you are restricted to only positive x values, so you can see how you can only pick a point from 0 to 60 and the rest is "inside" because you are constrained by x+y=60 and both being positive numbers.

Answer 17

oh i get ittt

Answer 18

then what if their was no restriction

Answer 19

If we could pick any negative number then nothing really would happen here. Try out some other numbers to see. Like, suppose we have -50=x then that means y=110 so that they both add up to make 60. However multiplying a negative by a positive won't change anything. The only problem that might occur is if both were negative somehow, then they could make a larger positive number. This happens sometimes, so watch out. Additionally,

Let's suppose we alter the question and say

x is no longer any positive number, it can only be between 10 and 20. Then you would have to check the points x=10 and x=20 because the derivative would tell you a point outside your domain, and x=20 and y=40 would of course be our answer.