# Find all numbers C that satisfy the conclusion of the MVT of f(x) = Xln(x) [1,2] ?

May 27, 2018

$c = \frac{4}{e}$

#### Explanation:

The mean value theorem states that there are numbers $c$ where

$f ' \left(c\right) = \frac{f \left(b\right) - f \left(a\right)}{b - a}$ if a function is continuous on $\left[a , b\right]$ and differentiable on $\left(a , b\right)$.

Let's do the math.

$f ' \left(c\right) = \frac{2 \ln \left(2\right) - 0}{2 - 1}$

$f ' \left(c\right) = 2 \ln 2$

We now set this equal to the derivative of $f \left(x\right)$ to solve for $c$.

$f ' \left(c\right) = \ln c + c \left(\frac{1}{c}\right) = \ln c + 1$

Therefore

$\ln c + 1 = 2 \ln 2$

$\ln c = 2 \ln 2 - 1$

$\ln c = \ln 4 - \ln e$

$\ln c = \ln \left(\frac{4}{e}\right)$

$c = \frac{4}{e}$

Hopefully this helps!