Curve Sketching $y=x^x$

$$\begin{array}{ll} x^2 &=9\\ x &=\pm 3 \end{array}$$

Harder example: (stationary points and curve sketching.

Consider the curve

$$y=x^x, \quad x>0$$

Find the coordinates of any stationary points and inflexions.

Sketch the curve.

[You may assume that $\lim_{x\rightarrow 0} x^x=1$. This can be proven by using logs and l’Hopital’s rule.]

See for example, 

https://www.youtube.com/watch?v=BCKp6Nfk3sg

Answers

$$y'=x^x(\ln x+1)$$
$$y''=x^x[(\ln x+1)^2+{1\over x}]$$
Stationary Point is $\left( 1/e,(1/e)^{1/e} \right) = approx (0.37,0.69) = approx (0.4,0.7)$ Minimum

Also note that $y''>0$ for $x>0$