Is every point of the constant function $f(x)=k$ where $k \in \mathbb{R}$ simultaneously a maximum, a minimum, and an inflection point?

Question

This topic came up whilst arguing with some colleagues about the following question:

True or false: if a continuous function f defined on closed interval $[a,b]$ then it has a maximum and a minimum.

Some of us said false and giving as a counter-example the constant function. However, some state that it is true.

I found a paper-ette by EHRHARD BEHRENDS, STEFAN GESCHKE, AND TOMASZ NATKANIEC, which supports the true side, however, nothing solid.

My counter argument is that if every point serves as a max, min, and inflection point simultaneously, it is equivalent with no point being one.

Which side is correct?