Question

Given the function `f(x) = x^3 + x - 1`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,4] and find the c?

Answer 1

Please see below.

Explanation:

You determine whether it satisfies the hypotheses by determining whether `f(x) = x^3+x-1` is continuous on the interval `[0,4]` and differentiable on the interval `(0,4)`.

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(4)-f(0))/(4-0)` on the interval `(0,4)`.

`f` is a polynomial function, so `f` is continuous on its domain, which includes `[0,4]`

`f'(x) = 3x^2+1` which exists for all `x` so it exists for all `x` in `(0,4)`

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

To find `c` solve the equation `f'(x) = (f(4)-f(0))/(4-0)`. Discard any solutions outside `(0,4)`.

You should get `c = (4sqrt3)/3`.