Question

Given the function `f(x)=(x^2-9)/(3x)`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

Answer 1

See below.

Explanation:

You determine whether it satisfies the hypotheses by determining whether `f(x) = (x^2-9)/(3x)` is continuous on the interval `[1,4]` and differentiable on the interval `(1,4)`.

You find the `c` mentioned in the conclusion of the theorem by solving `f'(x) = (f(4)-f(1))/(4-1)` on the interval `(1,4)`.

Answers

`f` is continuous on its domain, which includes `[1,4]`

`f'(x) = (x^2+9)/(3x^2)` which exists for all `x != 0` so it exists for all `x` in `(1,4)`

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

To find `c` solve the equation `f'(x) = (f(4)-f(1))/(4-1)`. Discard any solutions outside `(1,4)`.

I believe that you should get `c = 2` (because `-2` is outside the interval.).