Find extrema and monotonicity intervals of $ x^{x^2} $

Question

I'm trying to find minimum, maximum and intervals of monotonicity of following function:

$ f(x) = x^{x^2} $

I tried calculating derivatives, but they get very, very complicated really fast. How to approach this problem?

Answer 1

$$x^{x^2}=e^{x^2 \log x}\\ \frac d{dx}x^{x^2}=e^{x^2 \log x}(2x\log x + x)=x^{x^2}(2x \log x + x)$$ The $x^{x^2}$ is always positive, so to find zeros of the derivative we need $$2x \log x +x=0\\x(2\log x+1)=0\\ x=0,e^{-1/2}$$

Answer 2

The derivative of that function is $f´(x) = xx^{x^2}(2\log{x} + 1)$, which only zero is $x = \frac{1}{\sqrt{e}}$. The limit when $x$ tends to $0$ is $1$, and the function goes to infinity when $x$ tends to infinity, and you can see $f´(x) < 0 $ when $x \in (0,\frac{1}{\sqrt{e}})$ and $f´(x) > 0$ when $x> \frac{1}{\sqrt{e}}$. Then $x=\frac{1}{\sqrt{e}}$ is the only minimum.