Question

If the sum of two numbers is 3, then the maximum value of the product of the first number and the square of the second number is

(A) 4 

(B) 3 

(C) 2 

(D) 1

Answer 1

Answer is (A) 4

Let the first number be 3 − x and the second number be x.

Accordingly, we have to maximize (3 - x)x².

Let us consider

f(x) = (3 - x)x² = 3x² - x³  ⇒ f'(x) = 6x - 3x²

Therefore,

f'(x) = 0 ⇒ x = 0, 2

Also,

f''(x) = 6 - 6x

Obviously, f''(2) = - 6 < 0. Therefore, the required maximum value is (3 - 2)2² = 4.