If the function `f(x)=2x^3-9ax^2+12a^2x+1,` where `a gt 0,` attains its maximum and minimum at `p and q,` respectively, such that `p^2=q,` then `a` equal to (a) `1` (b) `2` (c) `1/2` (d) `3`

Options

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

Answer

2

Explanation

`f(x)=2x^(3)-9ax^(2)+12a^(2)x+1`
`therefore f(X)=6x^(2)-18ax+12a^(2)and f(x)=12x-18a`
For maximum /minimum , `6x^(2)-18ax+12a^(2)=0`
or `x^(2)-3ax+2a^(2)=0`
or `(x-a)(x-2a)=0`
i.e x=a or x =2a
Now `f''(a)=12a=18a=6alt0`
Therefore f(X) is maximum at x=a and minimum at x=2a
Thus p=a and q =2a
Given that `p^(2) = q or a^(2)=2a or a (a-2) =0 or a=2`