Question

How do you determine if rolles theorem can be applied to `f(x) = 2x^3 - x^2 - 8x + 4` on the interval [-2,2] and if so how do you find all the values of c in the interval for which f'(c)=0?

Answer 1

Rolle's Theorem has three hypotheses:

H1 : `f` is continuous on the closed interval `[a,b]`

H2 : `f` is differentiable on the open interval `(a,b)`.

H3 : `f(a)=f(b)`

In this question, `f(x) = 2x^3-x^2-8x+4` , `a=-2` and `b=2`.

We can apply Rolle's Theorem if all 3 hypotheses are true.

So answer each question:

H1 : Is `f(x) = 2x^3-x^2-8x+4` continuous on the closed interval `[-2,2]`?

H2 : Is `f(x) = 2x^3-x^2-8x+4` differentiable on the openinterval `(-2,2)`?

H3 : `Is f(-2)=f(2)`?

If the answer to all three questions is yes, then Rolle's can be applied to this function on this interval.

To solve `f'(c) = 0`, find `f'(x)`, set it equal to `0` and solve the equation. (There may be more than one solution.)
Select, as `c`, any solutions in `(-2,2)`
(That is where Rolle's says there must be a solution. There may be more than one solution in the interval.)