Question

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

Answer 1

Please see below.

Explanation:

Hypothesis : `f` is continuous on `[1/2,2]`

True. `f` is a rational function, so `f` is continuous on its domain. The domain is all reals except `0` and `0` is not in `[1/2,2]`.

Hypothesis: `f` is differentiable on `(1/2,2)`

True. Remember that "differentiable" means "the derivative exists".
We have `f'(x)=-1/x^2`, so `f'(x)` exists for all reals except `0` and `0` is not in `(1/2,2)`.

The conclusion says that there is a `c` in `(1/2,2)` with

`f'(c) = (f(2)-f(1/2))/(2-(1/2))`.

To find those `c`'s, solve the equation. Discard solutions outside of `(1/2,2)`.

`-1/x^2 = -1` has solutions `x = +-1`.

So `c = 1`