Question

Given the function `f(x)=x/(x+9)`, 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

Please see below.

Explanation:

The Mean Value Theorem has two Hypotheses.
A hypothesis is satisfied if it is true.

H1: The function must be continuous on `[1,4]`.

So, ask your self where `f` is NOT continuous. Are there any numbers in `[1,4]` where `f` is not continuous?

H2: The function must be differentiable on `(1,4)`.

Find `f'` and determine whether there are any numbers in `(1,4)` where `f'` is not defined.

To (try to) find the `c` mentioned in the conclusion of the Mean Value Theorem,

set `f'(x) = (f(4)-f(1))/(4-1)` and solve the resulting equation.

Since the conclusion asserts the existence of a `c` in the interval `(1,4)` any solutions to the equation that are in the interval are values for `c`.