Here we examine what the second derivative tells us about the geometry of functions.

The graphs of two functions, and , both increasing on the given interval, are given below.
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We know that the sign of the derivative tells us whether a function is increasing or decreasing at some point. Likewise, the sign of the second derivative tells us whether is increasing or decreasing at . If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. This is summarized in a single theorem.

We summarize the consequences of this theorem in the table below:

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