How to prove that $e^x = 8x^3$ has only one solution in $[0,1]$? Prove that $e^x = 8x^3$ has only one solution in $[0,1]$. 
If we define $f(x) = e^x - 8x^3$ then by mean value theorem there exists at least one solution. But $f$ is not strictly decreasing/increasing. How do I continue?

$f$ is concave on $[0.1,1]$, so $f(0.1)>0$ and $f(1)<0$ imply that the root from $(0.1,1)$ is unique. To check that there is no root on $(0,0.1)$, note that $f'(x)$ is positive here and $f(0)>0$, $f(0.1)>0$.
