I am trying to minimize the function in the form of $f(x) = (1-a^x)^x$ where $0 < a < 1$ with respect to $x$ (for $x > 0$) and I am stuck!
Unfortunately the derivative is not nice enough to use the traditional method of setting it equal to zero.
Some quick plots show that the minimizer should be something around $-1 / \log(a)$ but not exactly that. (Indeed $x^* = - 1 / \log(a)$ is the minimizer of $1 - x a^x$ which approximates $f(x)$ if $a \ll 1$).
I appreciate if you someone can give me hints or ideas about how to minimize such a function?