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 `p` and `q` , respectively, such that `p^2=q ,` then `a` equal to 1 (b) 2 (c) `1/2` (d) 3

Options

A. 0
B. 1
C. 2
D. None of these

Answer

C

Explanation

Given `f(x)=2x^(3)-9ax^(2)+12a^(2)x+1` attains maximum and minimum at p and q respectively.
`therefore f'(p)=0, f'(q)=0`
`f''(p) lt 0 and f''(q) gt 0`
`"Now", f'(p)=0`
and f'(q)=0
`Rightarrow 6p^(2)-18ap+12a^(2)=0`
`and 6q^(2)-18aq+12a^(2)=0`
`Rightarrow p^(2)-3ap+2a^(2)=0`
`and q^(2)-3aq+2a^(2)=0`
`Rightarrow p=a,2a=q=a, 2a......(i)`
`"Now," f''(p) lt 0`
`Rightarrow 12p-18alt0`
`Rightarrow p lt (3)/(2)a.......(ii)`
`and f''(q) gt 0 Rightarrow 12q-18a gt 0`
`Rightarrow q gt (3)/(2) a.........(iii)`
`"Now" p^(2)=q`
`Rightarrow a^(2)=2a`
`Rightarrow a=0,2`
But for `a=0, f(x)=2x^(3)+1` which does not attains a maximum or minimum for any value of x. Hence a=2.