creating a graph given the following stipulations

Question

Hi I'm very lost on a practice question I was attempting to solve, I don't know how to start or finish the problem.

Create a graph of f(x) given that f(x) satisfies:

f(0) = 0 and f(5) = 0.

(0,0) and (5,0) are both relative maximum points.

f'(0) and f'(5) are undefined.

f"(x) > 0 on (-∞ ,0), (0,5), (5,∞ )

Please show me your end result, preferably with the steps, so I can do a couple more on my own and improve my mathematics.

Answer 1

1. $f(0)=0$ and $f(5)=0$ is straightforward: make sure your graph goes through $(0,0)$ and $(5,0)$.

2. This is also intuitive: make sure at the end that these two points are local maxima; they should be the peak of a "hill" of the graph.

3. Check your notes/textbook for examples of functions that are not differentiable at a point.

A good example is the absolute value function $g(x)=|x|$, which is not differentiable at zero due to the point.

4. This means the curvature is positive on those three intervals. Visually, your graph should look like three "smiles," with little peaks at the two points $(0,0)$ and $(5,0)$.