Question

Maximum value of x(1 - x)², when 0 ≤ x ≤ 2, is

(A) 2/27 

(B) 4/27 

(C) 5 

(D) 0

Answer 1

Answer is (B) 4/27

Given that

f(x) = x(1 - x)²

That is,

f(x) = x³ - 2x² + x

Now, f'(x) = 3x² - 4x + 1

Substituting f'(x) = 0, we have

3x² - 4x + 1 = 0

3x² - 3x - x + 1 = 0 ⇒ x = 1, 1/3

Now, f''(x) = 6x - 4

Therefore, f''(1) = 2 is positive and f''(1/3) = - 2  is negative. Hence, the maximum value is x = 1/3. The maximum is

f(1/3) = 4/27