Question

Calculus:

For problems 1 & 2 determine all the number(s) c which satisfy the conclusion of Rolle’s Theorem for the given function and interval.

Answer 1

Both
`f` and
`g` satisfy the conditions for Rolle's theorem, which then says there exists
`c\in[-1,3]` (for
`f`) such that
`f'(c)=0`, and
`c\in[-2,1` (for
`g`) such that
`g'(c)=0`.

1.

`f(x)=x^2-2x-8\implies f'(x)=2x-2`

`f'(c)=2c-2=0\implies c=1`

2.

`g(t)=2t-t^2-t^3=0\implies g'(t)=2-2t-3t^2`

`g'(c)=2-2c-3c^2=0\implies c=\frac{-1\pm\sqrt7}3`