Find two numbers whose sum is 10, if the product of one to the square of the other is to be a maximum.

Answer 1

Let x and y be the two numbers.

From the condition: x + y = 10

For this problem, I think we're assuming that x and y are positive real numbers.

Therefore, 0 < x,y < 10

We're trying to maximize xy²

Solve for x in the condition: x = 10 - y

Plug that into the equation you're maximize since we want the equation we're maximizing to only have a single variable.

Substituting in, we get that we're trying to maximize (10-y)y² on the interval (0,10)

We'll call this function f(y)

f(y) = (10-y)y² = -y³ + 10y²

Look at the endpoints, f(0) and f(10)

You also have to find the critical points

f '(y) = -3y² + 20y = 0

f '(y) = y(-3y + 20) = 0

y = 20/3

f(0) = 0

f(10) = 0

f(20/3) =4000/27

f(20/3) gives us a maximum value so y = 20/3 and x = 10 - y = 10/3