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. 2
C. 1
D. `1/2`

Answer

B

Explanation

Given `f(x)=2x^(3)-9ax^(2)+12a^(2)x+1`
`impliesf'(x)=6x^(2)-18ax+12a^(2)`
For maxima or minima put `f'(x)=0`
`implies6(x^(2)-3ax+2a^(2))=0`
`impliesx^(2)-3ax+2a^(2)=0`
`impliesx^(2)-2ax-ax+2a^(2)=0`
`impliesx(x-2a)-a(x-2a)=0`
`implies(x-a)(x-2a)=0`
`implies x=a,x=2a`
Now `f''(x)=12x-18a`
At `x=a, f''(x)=12a-18a=-6a(-ve) [ :' agt0]`
`:.f(x)` will be maximum at `x=a`
i.e. `p=a`
At `x=2a,f"(x)=24a-18a=6a(+ve)`
`:.f(x)` will be minimum at `x=2a` i.e. `q=2a`
Given `p^(2)=q`
`implies a^(2)=2aimpliesa=2`