Question

How do you verify that the function `f(x)=x^3 - 21x^2 + 80x + 2` satisfies Rolle's Theorem on the given interval [0,16] and then find all numbers c that satisfy the conclusion of Rolle's Theorem?

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)=x^3 - 21x^2 + 80x + 2` , `a=0` and `b=16`.

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

So answer each question:

H1 : Is `f(x)=x^3 - 21x^2 + 80x + 2` continuous on the closed interval `[0,16]`?

H2 : Is `f(x)=x^3 - 21x^2 + 80x + 2` differentiable on the open interval `(0,16)`?

H3 : Is `f(0)=f(16)`?

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 `(0,16)`
(That is where Rolle's says there must be a solution. There may be more than one solution in the interval.)

Arithmetic note:
`f(16) = 16(16^2)-21(16^2)+80(16)+2`

since `80 = 5*16`, we get:

`f(16) = 16(16^2)-21(16^2)+5*16 (16)+2`
so

`f(16) = 16(16^2)-21(16^2)+5*(16^2)+2 = 21(16^2)-21(16^2)+2=2`

Please don't waste any of your youth finding `21*(16^2)`.)