# Problem understanding question

• May 12th 2013, 10:15 PM
Paze
Problem understanding question
I have a problem:

1. Find two positive numbers whose sum is 300 and whose product is a maximum.

What I don't understand is that last constraint: ''is a maximum''. What do they mean by a product being a maximum? Thanks.
• May 12th 2013, 11:07 PM
agentmulder
Re: Problem understanding question
We have x + y = 300 and want to maximize x*y. From the first condition we get y = 300 - x. Now substitute.

x(300 - x) = -x^2 + 300x

This is a parabola that opens DOWNWARD therefore the maximum occurs at the vertex. The x co-ordinate of the vertex is given by

$x \ = \ -\frac{b}{2a} \ = \ -\frac{300}{-2}$

Can you continue?

:)
• May 12th 2013, 11:19 PM
Paze
Re: Problem understanding question
Ah, I understand now what they mean.

Can't I simply take the derivative of the substitution and set it to 0 to come to a conclusion? (That's what the class is all about at the moment).

As so:

$\frac{d}{dx}-x^2+300x=-2x+300\\\\-2x+300=0\\\\x=150$

Which would imply that the largest number is given by

$150\cdot(300-150)=\\\\150\cdot150$

? (btw can someone tell me why the first part of these spaces always has an indent?)

Just one more question if this all checks out: Don't I have some specific range as well? I can't seem to put my finger on that range.
• May 13th 2013, 12:14 AM
agentmulder
Re: Problem understanding question
Yes you can use derivative, wasn't sure if this was a calculus or pre-calculus question. The positive values for both x and y occur between the 2 roots of the parabola and nowhere else as a quick sketch will show so,

$0 \ < \ x \ < \ 300$

and

$0 \ < \ y \ < \ 300$

That range for y is a bit sneaky... if you pick x to range between 0 and 300 then y will range between 0 and 150 but i suppose the roles of x and y can be reversed without fear so technically you get both between 0 and 300.

:)

The indent may have something to do with your settings but i don't know what, i'm using the same code as you but i don't get that indent.

P.S. If you're going to use calculus you will need to prove x = 150 gives the maximum by using the 1st derivative test.
• May 13th 2013, 04:41 AM
Paze
Re: Problem understanding question
Quote:

Originally Posted by agentmulder
Yes you can use derivative, wasn't sure if this was a calculus or pre-calculus question. The positive values for both x and y occur between the 2 roots of the parabola and nowhere else as a quick sketch will show so,

$0 \ < \ x \ < \ 300$

and

$0 \ < \ y \ < \ 300$

That range for y is a bit sneaky... if you pick x to range between 0 and 300 then y will range between 0 and 150 but i suppose the roles of x and y can be reversed without fear so technically you get both between 0 and 300.

:)

The indent may have something to do with your settings but i don't know what, i'm using the same code as you but i don't get that indent.

P.S. If you're going to use calculus you will need to prove x = 150 gives the maximum by using the 1st derivative test.

By 'first derivative test' do you mean this?

$f'(149)>0 , f'(151)<0$ Therefore x=relative maxima.

?
• May 13th 2013, 10:28 PM
agentmulder
Re: Problem understanding question
Yes, you are correct.

:)
• May 14th 2013, 05:37 AM
Paze
Re: Problem understanding question
Quote:

Originally Posted by agentmulder
Yes, you are correct.

:)

Thank you very much!