This statement is false.
Consider, for example, the graph of f ( x ) = tan ( x ) {\displaystyle f(x)=\tan(x)} for -π/2 < x < π/2. Nowhere do we have f ′ ( x ) = sec ( x ) = 0 {\displaystyle \displaystyle f'(x)=\sec(x)=0} , but still the curve switches from concave down to concave up.
