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Let f(x)=x^(3)+ax^(2)+bx+c, where a, b, ...

Let `f(x)=x^(3)+ax^(2)+bx+c`, where a, b, c are real numbers. If `f(x)` has a local minimum at `x = 1` and a local maximum at `x=-1/3` and `f(2)=0`, then `int_(-1)^(1) f(x) dx` equals-

A

`14/3`

B

`(-14)/3`

C

`7/3`

D

`(-7)/3`

Text Solution

Verified by Experts

The correct Answer is:
B
`f'(x)=3(x^(2)-2/3x-1/3)=3x^(2)-2x-1`
`f(x)=x^(3)-x^(2)-x+lambda`
`f(2)=8-4-2+lambda=0 rArr lambda=-2`
`f(x)=x^(3)-x^(2)-x-2`
`underset(-1)overset(1)(int) f(x) dx=-2 underset(0)overset(1)(int) (x^(2)+2)dx =-2 (1/3 +2)=(-14)/3`
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