Question

How do I find the lower bound of a function?

Answer 1

See below.

Explanation:

First of all, let's get rid of infinities: a function can tend to `\pm\infty` either at the extreme points of its domain or because of some vertical asymptote.

So, you should first of all check

`lim_{x \to x_0} f(x)`

for every point `x_0` at the boundary of the domain. For example, if the domain is `(-\infty,\infty)`, you should check

`lim_{x \to \pm\infty} f(x)`

If the domain is like `\mathbb{R}\setminus{2}` you should check

`lim_{x \to \pm\infty} f(x),\qquad lim_{x \to 2^\pm} f(x)`

and so on. If any of these limits is `-\infty`, the function has no finite lower bound.

Else, you can check the derivative: when you set `f'(x)=0`, you will find points of maximum of minimum. For every `x` which solves `f'(x)=0`, you should compute `f''(x)`. If `f''(x)>0`, the point is indeed a minumum.

Now, in the most general case, you have a collection of points `x_1,...,x_n` such that

`f'(x_i)=0,\qquad f''(x_i)>0` for every `i=1,.., n`

Which means that they are all local minima of your function. The lower bound of the function, i.e. the global minimum, will be the smallest image of those points: you just need to compare

`f(x_1), ..., f(x_n)`, and choose the smallest one.