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A nonzero polynomial with real coefficie...

A nonzero polynomial with real coefficient has the property that `f(x)=f^(prime)(x)dot f^(prime prime)(x)dot` If `a` is the leading coefficient of `f(x),` then the value of `1/(2a)` is____

A

`1//3`

B

6

C

12

D

`1//18`

Text Solution

Verified by Experts

The correct Answer is:
D
Let degree of f(x) is n,
Degree of `f'(x)` is `n-1`
Degree of `f''(x)` is `(n-1)`
Hence `n=(n-1)+(n-2)=2n-3`
`therefore" "n=3`
`therefore" "f(x)=ax^(3)+bx^(2)+cx+d," "(ane0)`
`" "f'(x)=3ax^(2)+2bx+c`
`" "f''(x)=6ax+2b`
`rArr" "ax^(3)+bx^(2)+cx+d=(3ax^(2)+2bx+c)(6ax+2b)`
`rArr" "18a^(2)=a`
`rArr" "a=1//18`
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