If the function `f(x)=2x^3-9a x^2+12 x^2x+1,w h e r ea >0,` attains its maximum and minimum at `pa n dq` , respectively, such that `p^2=q ,` then `a` equal to 1 (b) 2 (c) `1/2` (d) 3

Options

A. 3
B. 1
C. 2
D. `1//2`

Answer

C

Explanation

Given ` f (x ) = 2x ^ 3 - 9 ax^ 2 + 12 a ^ 2 x + 1 `
` f' ( x ) = 6 x ^ 2 - 18 ax + 12 a ^ 2 `
Put f' (x ) = 0, for maximum or minimum.
`rArr 6 ( x ^ 2 - 3ax + 2a^ 2 ) = 0 `
` rArr x ^ 2 - 3ax + 2a^ 2 = 0 `
`rArr ( x - a) ( x - 2a ) = 0 `
` rArr x = a, x = 2a `
` rArr f'' ( x ) = 12 x - 18 a `
At ` x = a `,
` f'' (x) = 12 a - 18 a = -6 a `
` therefore f (x ) ` will be maximum at ` x = a ` i.e. p = a
At ` x = 2a, f'' ( x ) = 24 a- 18 a = 6 a `
` therefore f ( x ) ` will be minimum at ` x = 2a ` i.e. ` q = 2a `
Given , ` p^ 2= q `
`rArr a ^ 2 = 2a rArr a = 2 `