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Here we examine what the second derivative tells us about the geometry of
The graphs of two functions, and , both increasing on the given interval, are given
Let be a function differentiable on an open interval .
We say that the graph of is concave up on if , the derivative of , is
increasing on .
We say that the graph of is concave down on if , the derivative of , is
decreasing on .
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
Test for Concavity Let be an open interval.
If for all in , then the graph of is concave up on .
If for all in , then the graph of is concave down on .
We summarize the consequences of this theorem in the table below:
Let be a continuous function and suppose that:
for and .
for and .
Sketch a possible graph of .
Start by marking points in the domain where the
derivative changes sign and indicate intervals where is increasing and intervals is
decreasing. The function has a negative derivative from to . This means that is increasingdecreasing on this interval. The function has a positive derivative from to . This means that is
increasingdecreasing on this interval. Finally, The function has a negative derivative from to . This
means that is increasingdecreasing on this interval.
Now we should sketch the concavity: concave upconcave down when the second derivative is positive, concave upconcave down when the second derivative is negative.