The problem I am trying to solve is: Give an example of a function whose graph is increasing on $(0,\infty)$ and concave down on $(0,\infty)$ and which passes through the points $(1,1)$ and $(2,3)$.
I could not recall a general approach to a question like this, so I used the trial-and-error method. Some possible parent functions that came to mind were $f(x)=\ln(x), f(x)=-\dfrac{1}{x}$, and the piecewise function $f(x)= \begin{cases} 5 & \text{ if } x=0 \\ \sqrt{x} & \text{ if } x>0 \end{cases}$. All of these functions are increasing and concave down on $(0,\infty)$.
However. when I tried to manipulate these functions to satisfy the points $(1,1)$ and $(2,3)$, I was unsuccessful in coming up with the exact function. I considered finding the slope between these two points, but that would lead to a linear function which has zero concavity.
Any suggestions as to how to proceed with this problem are appreciated.
Best Answer
You can take the function $ f $ defined by $$f(x)=a - \frac bx$$
with $$f(1)=a - b =1$$ and $$f(2)=a - \frac b2 =3$$
to find that $$f(x)=5-\frac 4x$$ is defined at $ (0,+\infty) $ and satisfies the desired conditions.