Sketching graphs with prescribed monotonicity, sign and concavity

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Sketch:

• (a) the graph of a function $f(x)$ that is always decreasing, always positive and always concave up;
• (b) the graph of a function $f(x)$ that is negative before $x=-1$ and positive after; moreover concave down before $x=3$ and concave up after;
• (c) the graph of a function that is concave up for $-3<x<3$ and concave down elsewhere, always negative, increasing up to $x=-5$ and then decreasing.

Solution

(a) A curve above the $x$-axis that descends with an ever-gentler slope and upward concavity: for example a branch of the decreasing exponential $f(x)=e^{-x}$ (positive, decreasing, $f''>0$).

(b) The function crosses the $x$-axis at $x=-1$ passing from negative to positive; the concavity changes at $x=3$ (inflection), from concave down to concave up. Draw an increasing curve crossing $(-1,0)$, with an inflection at $x=3$.

(c) A graph entirely below the $x$-axis, with a relative maximum at $x=-5$ (first increasing, then decreasing) and two inflections at $x=-3$ and $x=3$ separating the concave/convex regions: concave down for $x<-3$, concave up for $-3<x<3$, concave down for $x>3$.

The solutions are not unique: any graph consistent with the sign, monotonicity and concavity conditions is acceptable.