# Thread: Finding Critical Numbers for a derivative with euler's number

1. ## Finding Critical Numbers for a derivative with euler's number

This is a problem I came across in my Calculus textbook: g(x)=e^(-x)+e^(3x)

I am asked to find the open intervals over which this function increases and decreases. The original function isn't undefined for any value of x so I proceeded to find the derivative: g'(x)=3e^(3x)-e^(-x)

For the critical numbers there are no values of x for which this function is undefined (where the original is defined) but there seems to be one value of x for which the derivative will be equal to zero.

This is an odd-numbered problem in the book, so I checked the website to see what the critical number would be. The website stated that g'(x)=3e^(3x)-e^(-x)=0
when x=-(1/4)ln(3).

However the website did not show the steps to arrive at such a result. Would anyone be so kind as to explain the steps it takes to arrive at this possible critical number?

This website: http://www.calcchat.com/book/Calculus-ETF-5e/ and the input section 4.3 # 17 is the problem I am dealing with.

2. ## Re: Finding Critical Numbers for a derivative with euler's number

$\displaystyle 3 e^{3 x}=e^{-x}$

$\displaystyle 3 e^{4 x}=1$

3. ## Re: Finding Critical Numbers for a derivative with euler's number

I understand how you set the equation up at first, but not how you arrived at 3e^(4x)=1 or how I would get from that to -(1/4)ln3 as an x-value that makes the derivative equal to zero.

4. ## Re: Finding Critical Numbers for a derivative with euler's number

$\displaystyle 3 e^{3 x}=e^{-x}$

multiply both sides by $\displaystyle e^x$ ...

$\displaystyle 3e^{3x} \cdot e^x = e^{-x} \cdot e^x$

$\displaystyle 3e^{3x+x} = e^{-x+x}$

$\displaystyle 3e^{4x} = e^0$

$\displaystyle 3 e^{4 x}=1$

solve for x ...

$\displaystyle e^{4x} = \frac{1}{3}$

$\displaystyle \ln(e^{4x}) = \ln\left(\frac{1}{3}\right)$

$\displaystyle 4x = \ln{1} - \ln{3}$

$\displaystyle 4x = -\ln{3}$

$\displaystyle x = -\frac{1}{4}\ln{3}$

5. ## Re: Finding Critical Numbers for a derivative with euler's number

Thank you skeeter. That was cool