Question

Question #3fe4c

Answer 1

`c=4`

Explanation:

`f'(x) = -16+4x = 0` at `x=4`

`f` is continuous and differentiable everywhere because it is a polynomial.

`f(3) = 29 = f(5)` so the hypotheses are satisfied.

Answer 2

`c=4`

Explanation:

First, state that all quadratic functions are everywhere continuous and differentiable.

Next, solve for f(3) and f(5); they have to be equal.

`f(3)=1-16(3)+2(3)^2=-29`

`f(5)=1-16(5)+2(5)^2=-29`

Therefore, Rolle's Theorem applies.

To find the value(s) of `c` that the problem is asking for, find the first derivative of the function:

`f`'`(x)=-16+4x`

From this, solve for `x` where `f`'`(x)` equals `0` on the interval [3,5].

`4x=16`
`x=4`

Therefore `c=4`