Intervals of Convex and Concave function

Question

Find the intervals where the function is convex and concave. $$f (x) = e^{2x} - 2e^x$$

I tried differentiating twice, and my answer is: concave when $x < \ln (1/2)$ and convex when $x > \ln (1/2)$. However the key says the other way around...

Answer 1

$f'(x) = 2e^{2x} - 2e^x \to f''(x) = 4e^{2x} - 2e^x$. $f$ is concave if $f''(x) < 0$, and is convex if $f''(x) > 0$. $f''(x) < 0 \iff 2e^x\left(2e^x - 1\right)<0 \iff 2e^x - 1 < 0 \iff e^x < \dfrac{1}{2} \iff x < \ln\left(\dfrac{1}{2}\right)=-\ln 2$