Question

Given the function `f(x)=sqrt(2-x)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-7,2] and find the c?

Answer 1

You determine whether it satisfies the hypotheses by determining whether `f(x) = sqrt(2-x)` is continuous on the interval `[-7,2]` and differentiable on the interval `(-7,2)`. (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(2)-f(-7))/(2-(-7))` on the interval `(-7,2)`.

`f` is continuous on its domain, which includes `[-7,2]`

`f'(x) = (-1)/(2sqrt(2-x))` which exists for all `x < 2` so it exists for all `x` in `(-7,2)`

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

To find `c` solve the equation `f'(x) = (f(2)-f(-7))/(2-(-7))`. Discard any solutions outside `(-7,2)`.

You should get `c = -1/4`.