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

Answer

C

Explanation

Given `f(x)=2x^(3)-9ax^(2)+12a^(2)x+1`
`implies f'(x)=6x^(2)-18ax+12a^(2)`
`f(x)` attains maximum and minimum at `p` and `q`
respectively.
`:.f'(p)=0, f'(q)=0`
and `f''(p)lt0` and `f''(q)gt0`
Consider `f'(p)=0` and `f'(q)=0`
`implies6p^(2)-18ap+12a^(2)=0`
and `6q^(2)-18aq+12a^(2)=0`
`impliesp^(2)-3ap+2a^(2)=0`
and `q^(2)-3aq+2a^(2)=0`
`impliesp=a,2a,q=a,2a`.............i
Now `f''(p)lt0`
`implies12p-18alt0impliesp lt 3/2a`............ii
and `f''(q)gt0implies12q-18agt0`
`impliesq gt 3/2 a`...............iii
From eqs i, ii and iii we get
`p=a,q=2a`
Now `p^(2)=q`
`impliesa^(2)=2aimpliesa=0,2`
But for `a=0, f(x)=2x^(3)+1` which does not attain a maximum or minimum for any value of `x`
Hence `a=2`