Calculating the maximum value of this function

Question

I was messing around in Desmos Graphing Calculator and I input this function (graph here): $$ f(x) = 2^{x-1.1^{x}} $$ I noticed it had a peak at around $(24.663, 18442.801)$, and I tried using the derivative of the function to get that value and I got this: $$ f'(x) = 2^{x-1.1^{x}} \cdot ln(2) \cdot (1 - 1.1^{x} \cdot ln(1.1)) $$ The thing is, I don't have any idea of how to get the value of $x$ when $f'(x)=0$. Do I have to use non-conventional methods to get the value of $x$?

Answer 1

You're doing fine. Take your derivative and set it to be equal to $0$: $$0=2^{x-1.1^x}\cdot\log2\cdot(1-1.1^x\cdot\log1.1)$$ and notice that only the last factor has a root. Solve the equation \begin{align} 1-1.1^x\cdot\log1.1&=0\\ 1.1^x\cdot\log1.1&=1\\ 1.1^x&=\frac1{\log1.1}\\ x\cdot{\log1.1}&=\log\left(\frac1{\log1.1}\right)\\ x&=\frac{-\log\log1.1}{\log1.1}\approx24.66283 \end{align} where $\log$ is the natural logarithm.