Concavity

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

The graphs of two functions, $f$ and $g$ , both increasing on the given interval, are given below.

Let $f$ be a function differentiable on an open interval $I$ .

We say that the graph of $f$ is concave up on $I$ if $f'$ , the derivative of $f$ , is increasing on $I$ .
We say that the graph of $f$ is concave down on $I$ if $f'$ , the derivative of $f$ , is decreasing on $I$ .

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 $f''(x)$ tells us whether $f'(x)$ is increasing or decreasing at $x$ . 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.

Test for Concavity Let $I$ be an open interval.

If $f''(x)>0$ for all $x$ in $I$ , then the graph of $f$ is concave up on $I$ .
If $f''(x)<0$ for all $x$ in $I$ , then the graph of $f$ is concave down on $I$ .

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

Let $f$ be a continuous function and suppose that:

$f'(x) > 0$ for $-1< x<1$ .
$f'(x) < 0$ for $-2< x<-1$ and $1<x<2$ .
$f''(x) > 0$ for $-2<x<0$ and $1<x< 2$ .
$f''(x) < 0$ for $0<x< 1$ .

Sketch a possible graph of $f$ .

Start by marking points in the domain where the derivative changes sign and indicate intervals where $f$ is increasing and intervals $f$ is decreasing. The function $f$ has a negative derivative from $-2$ to $x=\answer [given]{-1}$ . This means that $f$ is increasingdecreasing on this interval. The function $f$ has a positive derivative from $x=\answer [given]{-1}$ to $x=\answer [given]{1}$ . This means that $f$ is increasingdecreasing on this interval. Finally, The function $f$ has a negative derivative from $x=\answer [given]{1}$ to $2$ . This means that $f$ 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.

Finally, we can sketch our curve: