# Thread: Proving an inequality using the Mean Value Theorem

1. ## Proving an inequality using the Mean Value Theorem

I am trying to prove the following inequality using the Mean Value Theorem:

$\displaystyle \ln (3x) \leq 3x - 1$

I can do it other ways but the question says to use the Mean Value Theorem.

I have tried applying the Mean Value Theorem to the function $\displaystyle f(x) = \ln (3x)$ on the interval $\displaystyle \left[\frac{1}{3}, \, x\right]$ but cannot get the required inequality.

If anyone can help I would be grateful.

2. Apply the Mean Value Theorem to the function $\displaystyle f(x)=\ln(3x)-3x+1, \ x>0$ on the intervals $\displaystyle \left[\displaystyle\frac{1}{3},x\right]$ and $\displaystyle \left[ x,\displaystyle\frac{1}{3}\right]$.

3. Originally Posted by red_dog
Apply the Mean Value Theorem to the function $\displaystyle f(x)=\ln(3x)-3x+1, \ x>0$ on the intervals $\displaystyle \left[\displaystyle\frac{1}{3},x\right]$ and $\displaystyle \left[ x,\displaystyle\frac{1}{3}\right]$.