Question

Given the function `f(x)=x^2 - 2x + 5`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,3] and find the c?

Answer 1

You determine whether it satisfies the hypotheses by determining whether `f(x) = x^2-2x+5` is continuous on the interval `[1,3]` and differentiable on the interval `(1,3)`.

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(3)-f(1))/(3-1)` on the interval `(1,3)`.

So

`f` is continuous on its domain, which includes `[1,3]`

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

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

To find `c` solve the equation `f'(x) = (f(3)-f(1))/(3-1)`. Discard any solutions outside `(1,3)`.

You should get `c = 2`.