The Second Derivative and the Shape of a Function f(x)

The first derivative of a function gives information about the shape of the function, so the second derivative of a function gives information about the shape of the first derivative and about the shape of the function. In this section we investigate how to use the second derivative and the shape of the first derivative to reach conclusions about the shape of the function. The first derivative tells us whether the graph of $\mathrm{f}$ is increasing or decreasing. The second derivative will tell us about the "concavity" of $\mathrm{f}$, whether $\mathrm{f}$ is curving upward or downward.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.4-Second-Derivative.pdf
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