Question

Given the function `f(x)=x/(x+6)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,1] and find the c?

Answer 1

See below.

Explanation:

You determine whether it satisfies the hypotheses by determining whether `f(x) = x/(x+6)` is continuous on the interval `[0,1]` and differentiable on the interval `(0,1)`.

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(1)-f(0))/(1-0)` on the interval `(0,1)`.

`f` is continuous on its domain, which includes `[0,4]`

`f'(x) = 6/(x+6)^2` which exists for all `x != -6` so it exists for all `x` in `(0,1)`

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(0))/(1-0)`. Discard any solutions outside `(0,1)`.

You should get `c = -6+sqrt42`.