Some Calculus I homework help

The sum of two positive numbers is 75. Exactly what are the numbers if the product of one number times the cube of the other is to be maximum?

How would I go about doing this?

Hello,

Well, the way to do that is to learn to read...

"two positive numbers" ► x and y
"The sum of two positive numbers" ► x + y
"The sum of two positive numbers is 75" ► x + y = 75

"one number" ► x
"the other" ► y
"the cube of the other" ► y³
"the product of one number times the cube of the other" ► xy³
"the product of one number times the cube of the other is to be maximum" ► xy³ = MAX

So you have to solve the system:
{ x + y = 75
{ xy³ = MAX

x = 75 - y
xy³ = (75 - y)y³ = 75y³ - y⁴

f(y) = 75y³ - y⁴
f'(y) = 225y² - 4y³ = y²(225 - 4y)
f'(y) = 0    if y=0 or y=225/4

Hence the maximum is reached when y=225/4. Then x=75-y=75/4.
The maximum value is then 854296875/256.

The solution is then:
{ x = 75/4
{ y = 225/4

Regards,
Dragon.Jade :-)

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