Differentiability of $f’$ and inflection points

Solution 1:

The correct definition of inflection point is a point where concavity of the function changes. This may well happen at a point where the function is not differentiable. Consider, for example, $$f(x) = \begin{cases} x^2, & x<0 \\ \sqrt x, & x\ge 0. \end{cases}$$ This function is concave up for $x<0$ and concave down for $x>0$, and so $0$ is an inflection point, even though $f'(0)$ does not exist.

Many calculus books lead readers to think that we must have second derivatives changing from positive to negative (or vice versa) in order to test for inflection points; even worse, some readers think that any point with zero second derivative must be an inflection point. (Consider $f(x)=x^4$ at the origin.)