Increasing/Decreasing Test with Exponential Function

Question

The goal is to find the intervals by which the function $f(x) = e^{x} - e^{2x}$ is increasing and decreasing, as well as any local maxima/minima, intervals of concavity, inflection points, asymptotes, etc. with the intent to sketch the function. I've found the first derivative to be $f'(x) = e^{x} - 2e^{2x}$. To find the intervals, I'm attempting to set the derivative equal to zero and then find changes in sign, and this is where I'm getting stuck. Can the function ever be zero? How do I find the interval if it cannot be? Thanks in advance.

Answer 1

If you set $y=e^x$, your derivative is $y-2y^2$. You can't have $y=0$, but there is another point where the derivative is zero.

Answer 2

$e^x - 2e^{2x}=0$

$e^x(1-2e^x)=0$

Note that $e^x$ can never equal $0$, so set $1-2e^x$ equal $0$.

$1-2e^x=0$

$2e^x = 1$

$e^x = \frac{1}{2}$

$ln(e^x) = ln(\frac{1}{2})$

$x = ln(\frac{1}{2})$