What if we have 2 same local extrema, how do we deal with global extrema?

Question

I wonder, if I have to analyze some function and find global extrema.

for example this function:

It has 2 equal local maxima. So what local maximum should I settle as global? Can I both or have to pick one? I don't know. Will you help me with this?

Answer 1

A direct consequence of the defintion of global maxima is that the global maximum is necessarily unique, for if two real numbers $x,\ y$ are distinct, then by the order on $\mathbb{R}$ it must be the case that one is bigger than the other. What it can happen is that there are multiple, even an infinite amount of argmax, i.e. points whose image through a function gives the function's maximum. Therefore

$$f:\mathbb{R}\to\mathbb{R}:x\mapsto\sqrt{x^2-x^4}$$ $$\Rightarrow\max f= \frac{1}{2}, \text{argmax} f=\{-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\}$$