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,
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 $x>0$
