Question

The sum of the first number cubed and the second number is 500 and the product is a maximum.

Answer 1

We are given the following information:

the sum of the first number cubed and the second number is 500

their product is a maximum

We are looking for the 2 missing numbers.

To answer this, let's represent the two numbers as x and w.

From the given, we can form the following equation:

$x^3+w=500$

We can then express y as:

$w=500-x^3$

We can express their product as:

$x(500-x^3)=500x-x^4$

To find the maximum value of x, let's solve for the derivative of -x^4 + 500 x.

$\begin{gathered} f(x)=-x^4+500x \\ f^{\prime}(x)=-4x^3+500 \end{gathered}$

Then we solve for the value of x where f'(x) = 0.

$\begin{gathered} -4x^3+500=0 \\ -4x^3=-500 \\ x^3=125 \\ x=5 \end{gathered}$

Then we use x = 5 to solve for the second number, w.

$\begin{gathered} w=500-x^3 \\ w=500-5^3 \\ w=500-125 \\ w=375 \end{gathered}$

Therefore, the two numbers are 5 and 375.