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:
[tex]x^3+w=500[/tex]
We can then express y as:
[tex]w=500-x^3[/tex]
We can express their product as:
[tex]x(500-x^3)=500x-x^4[/tex]
To find the maximum value of x, let's solve for the derivative of -x^4 + 500 x.
[tex]\begin{gathered} f(x)=-x^4+500x \\ f^{\prime}(x)=-4x^3+500 \end{gathered}[/tex]
Then we solve for the value of x where f'(x) = 0.
[tex]\begin{gathered} -4x^3+500=0 \\ -4x^3=-500 \\ x^3=125 \\ x=5 \end{gathered}[/tex]
Then we use x = 5 to solve for the second number, w.
[tex]\begin{gathered} w=500-x^3 \\ w=500-5^3 \\ w=500-125 \\ w=375 \end{gathered}[/tex]
Therefore, the two numbers are 5 and 375.
