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

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.

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: