Concavity

Let $f$ be a twice differentiable function on an open interval $I.$

• If $f''(x)>0$ for all $x$ in $I,$ the graph of $f$ is concave upward.
• If $f''(x)<0$ for all $x$ in $I,$ the graph of $f$ is concave downward.
• If the sign of $f''(x)$ changes on either side of $c,$ then the point $(c ,f(c))$ is called a point of inflection. Furthermore, either $f''(c)=0$ or $f''(c)$ is undefined.