Proving an inequality using the Mean Value Theorem

**strugglingstudent1** · August 21st 2010, 07:54 PM

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

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

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 $f(x) = \ln (3x)$ on the interval $\left[\frac{1}{3}, \, x\right]$ but cannot get the required inequality.

If anyone can help I would be grateful.

**red_dog** · August 21st 2010, 11:59 PM

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

**strugglingstudent1** · November 8th 2010, 04:02 PM

Originally Posted by red_dog

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

Thankyou, very helpful.

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