The first derivative can be used to determine when a function is increasing and when it is decreasing. However, a function can be increasing (or, respectively, decreasing) and be concave up or it can be increasing (or, respectively, decreasing) and be concave down but the fist derivative cannot distinguish between these possibilities. As we shall see the second derivative can.


Geometrically, a function said to be concave up (or convex down) on an interval if its curve bends upward and concave down on an interval if its curve bends downward. In note that each curve lies above the tangent lines whereas in, each curve lies below the tangent lines.

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