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: