Let us consider $f(x) = x |x| - x.$ Find the critical and inflection points.
I have found the critical points by setting $f'(x) = 0.$ It gives $-\frac12, 0, \frac12$ are the critical points. But to find inflection points I set $f''(x) = 0$ and $x = 0$ becomes an inflection point. But graph of this function says that $-\frac12$ and $\frac12$ are also inflection points as $f(x)$ changes sign there. Kindly tell me about this.