The construction of functions given set of properties

Problem

Give the term and the domain of a function which fulfills the given property/properties.

1. The point (2|0) is a point of inflection of the graph of \(g\).

2. The graph of the function \(h\) is strictly monotonically decreasing and concave.

Solution of part a

A function with the desired point of inflection can be obtained from the following requirements:

\[\begin{split}f(2) &= 0\\ f''(2) &= 0\end{split}\]

Furthermore, we require that \(f'''(2)\neq0\) and choose in particular \(f'''(2)=1\). Taking these requirements into account, the integration yields

\[f(x) = \frac{1}{6}(x-2)^3+c(x-2)\,.\]

A plot made by Sage can confirm the point of inflection:

Solution of part b

The simplest solution for a strictly monotonically increasing and convex function is the exponential function. With a negative sign, it is turned into a strictly monotonically decreasing, concave function as can be confirmed by calculation:

\[\begin{split}h(x) &= -e^x \\ h'(x) &= -e^x < 0 \\ h''(x) &= -e^x < 0\end{split}\]

A drawing by Sage confirms this: