CPM Homework Help : APCALC Problem 7-11

Find two positive numbers whose sum is $8$ , such that the sum of the square of one number and the cube of the other number is a minimum. Use calculus to justify your solution.

Step 1:

Let $y = a^2 + b^3$
Use the fact that $a + b = 8$ to rewrite $y$ with only one variable.

Step 2:

Optimize! Find $y^\prime$ and set it equal to $0$ .

Hint:

Be sure to answer the question completely. In other words: What is $a$ ? What is $b$ ?