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Here we examine what the second derivative tells us about the geometry of
So far, we know how to determine if a function is increasing or decreasing by looking
at the sign of its derivative. If all we know about is the increasing/decreasing
information, we no not have enough information to say how is behaving accurately
enough. Consider the following four possibilities:
In both graphs in the left-hand column, is decreasing. In both graphs in the
right-hand column, is increasing. What is the difference between the two
rows? Their concavity.
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
That is, the graph of is concave up if the graph locally lies above its tangent lines,
and is concave down if the graph locally lies below its tangent lines.
The graphs of two functions, and , both increasing on the given interval, are given
A graph of is given below, with domain .
On what intervals is concave up? On what intervals is concave down?
some tangent lines on the graph.
The graph is above the tangent lines for in , and below the tangent lines we sketched
Therefore, is concave up on , and concave down on .
In that example, examine the intervals where was concave up. At the beginning of
that interval, was decreasing, and at the end, was increasing. The slopes in that
interval increased from a negative value to a positive value. In the interval where was
concave down, the slopes started positive and ended negative. The slopes in that
interval were decreasing.
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 . Let’s use this to add more details into the chart from
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. It is worth summarizing what we have seen already in to 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 .
Let be a continuous function and suppose that:
for and .
for and .
Sketch a possible graph of .
Start by marking 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.