Question

Find two positive numbers x and y such that their sum is 35 and the product x²y⁵ is a maximum

Answer 1

Let one number be x. Then, the other number is y = (35 − x).

Let P(x) = x²y⁵. Then, we have:

x = 0, x = 35, x = 10

When x = 35, and y = 35 − 35 = 0. This will make the product x² y⁵ equal to 0.

When x = 0, y = 35 − 0 = 35 and the product x²y² will be 0.

∴ x = 0 and x = 35 cannot be the possible values of x.

When x = 10, we have:

∴ By second derivative test, P(x) will be the maximum when x = 10 and y = 35 − 10 = 25.

Hence, the required numbers are 10 and 25.