Sketching a function with prescribed sign and concavity

Text

Sketch the graph of a function $f(x)$ that is positive before $x=-1$ and negative after; moreover the graph must be concave up before $x=3$ and concave down after.

Solution

The conditions impose two notable points:

• sign-changing zero at $x=-1$: the curve crosses the $x$-axis at $(-1,0)$ passing from positive to negative (so it is decreasing there);
• inflection at $x=3$: the concavity changes from upward ($f''>0$ for $x<3$) to downward ($f''<0$ for $x>3$).

One possible graph: it starts at the upper left above the $x$-axis, descends crossing $(-1,0)$, continues in the negative half-plane staying concave up until $x=3$, where it has an inflection and becomes concave down. The solution is not unique.