A function where absolute maximum is also absolute minimum?

Question

What is an example of a real-valued function where an absolute maximum is also an absolute minimum?

Answer 1

Hint. Suppose that the absolute maximum value is $M$. By definition this means that $f(x)\le M$ for every value of $x$. If the same $M$ is the absolute minimum value, this means that. . .

Can you finish this and determine what $f(x)$ is?

Answer 2

$$\large{\large{\large{\large{\large{-}}}}}$$

Answer 3

If $a$ is such that $f(a) \leq f(x)$ and $f(x) \leq f(a)$ for all $x$ in the domain of $f$ we have $f(a) \leq f(x) \leq f(a)$ which implies $f(a) = f(x)$, that is $f$ is constant.

Answer 4

The only functions which have this property are the constant functions.

Why?

Suppose $f(x) \geq f(m)$ for all $x$. Then $x=m$ is the global minimum of $f$. Likewise, if $f(x) \leq f(M)$ for all $x$. Then $x=M$ is the global maximum of $f$. So if $f(M)=f(m)$, we have $f(m) \leq f(x) \leq f(M)=f(m)$ for all $x$. This forces $f(x)=f(m)=f(M)$ for all $x$ so that $f$ is constant.