Concavity

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

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 $x$ . We summarize the consequences of this seemingly simple idea in the table below:

Inflection points

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 Suppose that $f''(x)$ exists on an interval.

(a): If $f''(x)>0$ on an interval, then $f$ is concave up on that interval.
(b): If $f''(x)<0$ on an interval, then $f$ is concave down on that interval.

Of particular interest are points at which the concavity changes from up to down or down to up.

If $f$ is continuous and its concavity changes either from up to down or down to up at $x=a$ , then $f$ has an inflection point at $x=a$ .

It is instructive to see some examples of inflection points:

It is also instructive to see some nonexamples of inflection points:

We identify inflection points by first finding $x$ such that $f''(x)$ is zero or undefined and then checking to see whether $f''(x)$ does in fact go from positive to negative or negative to positive at these points.

Even if $f''(a) = 0$ , the point determined by $x=a$ might not be an inflection point.

Describe the concavity of $f(x)=x^3-x$ .

To start, compute the first and second derivative of $f(x)$ with respect to $x$ , $f'(x)=\answer [given]{3x^2-1}\qquad \text {and}\qquad f''(x)=\answer [given]{6x}.$ Since $f''(0)=0$ , there is potentially an inflection point at $x=0$ . Using test points, we note the concavity does change from down to up, hence there is an inflection point at $x=0$ . The curve is concave down for all $x<0$ and concave up for all $x>0$ , see the graphs of $f(x) = x^3-x$ and $f''(x) = 6x$ .

Note that we need to compute and analyze the second derivative to understand concavity, so we may as well try to use the second derivative test for maxima and minima. If for some reason this fails we can then try one of the other tests.

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:

Graphs of higher order derivatives

Below we have graphed $y=f(x)$ :

Is the first derivative positive or negative on the interval $-1<x<1/2$ ?

Positive Negative

Below we have graphed $y=f'(x)$ :

Is the graph of $f(x)$ increasing or decreasing as $x$ increases on the interval $-1<x<0$ ?

Increasing Decreasing

Here we have unlabeled graphs of $f$ , $f'$ , and $f''$ :

Identify each curve above as a graph of $f$ , $f'$ , or $f''$ .

Here we see three curves, $A$ , $B$ , and $C$ . Since $A$ is positive negative increasing decreasing when $B$ is positive and positivenegativeincreasingdecreasing when $B$ is negative, we see $A'=B.$ Since $B$ is increasing when $C$ is positivenegativeincreasing decreasing and decreasing when $C$ is positivenegativeincreasingdecreasing , we see $B'=C.$ Hence $f=A$ , $f'=B$ , and $f''=C$ .

Here we have unlabeled graphs of $f$ , $f'$ , and $f''$ :

Identify each curve above as a graph of $f$ , $f'$ , or $f''$ .

Here we see three curves, $A$ , $B$ , and $C$ . Since $B$ is positivenegativeincreasingdecreasing when $A$ is positive and positivenegativeincreasingdecreasing when $A$ is negative, we see $B'=A.$ Since $A$ is increasing when $C$ is positivenegativeincreasingdecreasing and decreasing when $C$ is positivenegativeincreasingdecreasing , we see $A'=C.$ Hence $f=\answer [given]{B}$ , $f'=\answer [given]{A}$ , and $f''=\answer [given]{C}$ .

Here we have unlabeled graphs of $f$ , $f'$ , and $f''$ :

Identify each curve above as a graph of $f$ , $f'$ , or $f''$ .

Here we see three curves, $A$ , $B$ , and $C$ . Since $C$ is positivenegativeincreasingdecreasing when $B$ is positive and positivenegativeincreasingdecreasing when $B$ is negative, we see $C'=B.$ Since $B$ is increasing when $A$ is positivenegativeincreasingdecreasing and decreasing when $A$ is positivenegativeincreasingdecreasing , we see $B'=A.$ Hence $f=\answer [given]{C}$ , $f'=\answer [given]{B}$ , and $f''=\answer [given]{A}$ .