Question

Given the function `f(x)=5sqrt(25-x^2)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,5] and find the c?

Answer 1

You determine whether it satisfies the hypotheses by determining whether `f(x) = 5sqrt(25-x^2)` is continuous on the interval `[0,5]` and differentiable on the interval `(0,5)`.
(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(5)-f(0))/(5-0)` on the interval `(0,5)`.

`f` is continuous on its domain, which includes `[0,5]`

`f'(x) = (- 5x)/sqrt(25-x^2)` which exists for all `x` in `(-5,5)` so it exists for all `x` in `(0,5)`

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

To find `c` solve the equation `f'(x) = (f(5)-f(0))/(5-0)`. Discard any solutions outside `(0,5)`.

You should get `c = (5sqrt2)/2`. (The negative solution is not in the interval.)