f(x) is a polynomial function, `f: R rarr R,` such that `f(2x)=f'(x)f''(x).` Equation f(x) = x has (A) three real and positive roots (B) three real and negative roots (C) one real root (D) three real roots such that sum of roots is zero

Options

A. three real and positive roots
B. three real and negative roots
C. one real root
D. three real roots such that sum of roots is zero

Explanation

Suppose degree of f(x) = n. Then degree of f'=n-1 and degree of `f''=n-2.` So,
`n=n-1+n-2`
Hence, n=3.
So, put `f(x)=ax^(3)+bx^(2)+cx+d (where a ne 0).`
From `f(2x)=f'(x).f''(x),` we have
`8ax^(3)+4bx^(2)+2cx+d=(3ax^(2)+2bx+c)(6ax+2b)`
`=18a^(2)x^(3)+18abx^(2)+(6ac+4b^(2))x+2bc`
Comparing coefficients of terms, we have
`18a^(2)=8a" "rArra=4//9`
`18ab=4b" "rArrb=0`
`2c=6ac+4b^(2)" "rArrc=0`
`d=2bc" "rArrd=0`
`"or "f(x)=(4x^(3))/(9),` which is clearly one-one and onto
`"or "f(3)=12`
`"Also, "(4x^(3))/(9)=xor x=0, x= pm 3//2.`
Hence, sum of roots of equation is zero.