It is given that the Rolles theorem holds for the function `f(x)=x^3+b x^2+c x ,\ \ x in [1,2]` at the point `x=4/3` . Find the values of `b` and `c` .

Options

A. `b=8, c=-5`
B. `b=-5, c=8`
C. `b=5, c=-8`
D. `b=-5, c=-8`

Explanation

It is given that the Rollle's theorem holds for `f(x)=x^(3)+bx^(2)+cx` on [1,2]
`:. f(1)=f(2) and f'((4)/(3))=0`
`rArr 1+b+c=8+4b+2c`
and `,3((4)/(2))^(2)+2b((4)/(3))+c=0 " " [ :. f'(x)=3x^(2)=3x^(2)+2 bx+c]`
`rArr 3b+c+7=0 and 8b+3c+16=0`
gt `rArr b=-5, c=8`