Second-order condition

Introduction: From the graph of a convex function $y(x)$ it follows that the slope of the tangent line to the graph increases with $x$. Hence, a function $y(x)$ is convex if the derivative $y'(x)$ increases. Note that the derivative $y'(x)$ increases if $y''(x)\geq 0$.
Similarly, it follows that the function $y(x)$ is concave if the derivative $y'(x)$ decreases.

Second-order condition for a convex/concave function

• If $y''(x)\geq 0$ on an interval, then the function $y(x)$ is convex on that interval.
• If $y''(x)\leq 0$ on an interval, then the function $y(x)$ is concave on that interval.