Ask Uncle Colin: A Pointy Thing

Dear Uncle Colin,

What do you call the ‘point’ of the absolute value graph - for instance, $(0,0)$ on the graph of $y=|x|$? It can’t be a minimum because the gradient is undefined!

Proof Or It’s Not True

Hi, POINT, and thanks for your message!

I hate to go ‘well, actually’, but I’m afraid I have to here: the name of the point you’re referring to is indeed a (global and local) minimum. It’s just not a minimum stationary point.

(Throughout this post, I’ll talk about minima - the same reasoning, with obvious changes, goes for maxima).

So what is a minimum?

A minimum is simply the least value obtained by a function. There are two varieties: a global minimum is the smallest value the function has over the whole of its domain; a local minimum is the smallest value in its neighbourhood (that is, you can choose an interval $[a,b]$ such that $a<x_0<b$, and $f(x)\ge f(x_0)$ for every $x$ in the interval.)

Often - at least in the kind of functions mathematicians tend to look at - a minimum of either flavour corresponds to a stationary point on the graph. That’s somewhere the function’s first derivative is 0 and the second derivative is positive ¹.

But that’s not always the case: for example, the function $f(x)=\frac{1}{x}$, $0<x\le 100$ has a global minimum value of $\frac{1}{100}$. Nowhere on the graph has a lesser $y$-value.

Infimum

It’s also worth mentioning (since we’re here), the idea of the infimum: sometimes a function will descend as close as you like to a value without ever reaching it. For example, it would be nice to look at $f(x)=\frac{1}{x}$ for $x>0$ and say ‘that has a minimum value of 0’ - but it never reaches zero!

Instead, this function has an infimum of zero - it’s the largest value $I$ such that $f(x)\ge I$ for all $x$ in the domain.

Hope that helps!

- Uncle Colin

Footnotes:

1. or rather, the first non-zero derivate is an even-numbered derivative, and has a positive value