Concavity

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

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 $f$ is the increasing/decreasing information, we no not have enough information to say how $f$ is behaving accurately enough. Consider the following four possibilities:

In both graphs in the left-hand column, $f$ is decreasing. In both graphs in the right-hand column, $f$ is increasing. What is the difference between the two rows? Their concavity.

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 $I$ .

That is, the graph of $f$ 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, $f$ and $g$ , both increasing on the given interval, are given below.

A graph of $y=f(x)$ is given below, with domain $\left (-\frac {3}{2} , \frac {3}{2} \right )$ .

On what intervals is $f$ concave up? On what intervals is $f$ concave down?

Let’s draw some tangent lines on the graph.

The graph is above the tangent lines for $x$ in $\left (-\frac {3}{2}, 0\right )$ , and below the tangent lines we sketched on $\left (0, \frac {3}{2}\right )$ .

Therefore, $f$ is concave up on $\left (\answer {-\frac {3}{2}}, \answer {0}\right )$ , and concave down on $\left (\answer {0}, \answer {\frac {3}{2}}\right )$ .

In that example, examine the intervals where $f$ was concave up. At the beginning of that interval, $f$ was decreasing, and at the end, $f$ was increasing. The slopes in that interval increased from a negative value to a positive value. In the interval where $f$ 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 $f''(x)$ tells us whether $f'(x)$ is increasing or decreasing at $x$ . Let’s use this to add more details into the chart from above.

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 theorem.

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

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

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 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: