Question

Is it true that any even function has either a maximum value or a minimum value?

Answer 1

False

Explanation:

Consider the function:

`f(x) = x sin x`

Note that this is defined for any `x in RR` and is an even function:

`f(-x) = (-x) sin(-x) = x sin x = f(x)" "` for all `x in RR`

It is unbounded and taking arbitrarily large positive and negative values as `x->oo` and `x->-oo`.

Its graph looks like this:

graph{x sin x [-20, 20, -10, 10]}

So it has no global minimum or maximum.

What about local maxima or minima?

Again, false.

If `c(x)` is Conway's base 13 function (https://en.wikipedia.org/wiki/Conway_base_13_function) then we can define an even function:

`f(x) = c(abs(x))`

Conway's base 13 function is discontinuous everywhere and takes every real value in any interval - so no local minimum or maximum.