Question

Given the function `f(x) = x² - 3x + 1`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,1] and find the c?

Answer 1

See below.

Explanation:

You determine whether it satisfies the hypotheses by determining whether `f(x) =x^2-3x+1` is continuous on the interval `[-1,1]` and differentiable on the interval `(-1,1)`. (Those are the hypotheses of the Mean Value Theorem)

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(1)-f(-1))/(1-(-1))` on the interval `(-1,1)`.

`f` is a polynomial, so it is continuous on its domain, which includes `[-1,1]`

`f'(x) = 2x-3` which exists for all `x` so it exists for all `x` in `(-1,1)`

Therefore this function satisfies the hypotheses of the Mean Value Theorem on this interval.

To find `c` solve the equation `f'(x) = (f(1)-f(-1))/(1-(-1))`. Discard any solutions outside `(-1,1)`.

You should get `c = 0`.