# 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?

Dec 11, 2016

See below.

#### Explanation:

You determine whether it satisfies the hypotheses by determining whether $f \left(x\right) = \frac{x}{x + 6}$ is continuous on the interval $\left[0 , 1\right]$ and differentiable on the interval $\left(0 , 1\right)$.

You find the $c$ mentioned in the conclusion of the theorem by solving $f ' \left(x\right) = \frac{f \left(1\right) - f \left(0\right)}{1 - 0}$ on the interval $\left(0 , 1\right)$.

$f$ is continuous on its domain, which includes $\left[0 , 4\right]$

$f ' \left(x\right) = \frac{6}{x + 6} ^ 2$ which exists for all $x \ne - 6$ so it exists for all $x$ in $\left(0 , 1\right)$

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

To find $c$ solve the equation $f ' \left(x\right) = \frac{f \left(1\right) - f \left(0\right)}{1 - 0}$. Discard any solutions outside $\left(0 , 1\right)$.

You should get $c = - 6 + \sqrt{42}$.