Application of Derivatives · Question 18

If the function f(x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p2 = q, then f(3) is equal to :

1. 55
2. 37
3. 10
4. 23

Answer

(2) 37

Solution

f'(x) = 6x2 - 18ax + 12a2 = 6(x - a)(x - 2a)

f''(x) = 12x - 18a f''(a) = -6a < 0 and f''(2a) = 6a > 0 ( a > 0)

Local maximum at p = a, local minimum at q = 2a

p2 = q a2 = 2a a = 2

f(x) = 2x3 - 18x2 + 48x + 1

f(3) = 54 - 162 + 144 + 1 = 37

Answer: (2)