Open critical points

Question

If an open critical point would have otherwise been an absolute max, why can’t a closed point next to it be an absolute max? In other words, why does an absolute max have to be a critical point or end point? It seems an arbitrary requirement. Is it for the sake of calculation?

Answer 1

let's ask the question

given f(x)=x² for x ∈ [-2,3)

and f(x)=0 for x=3

what is the maximum value of f(x)?

we know the limit by limit theorems at of f(x)=x² as x approaches 3, x increasing

lim x->3- for f(x)=9

limit of f(x) is 9 as f(x) approaches x=3

that is, for any small number b, |f(x)-9|<b for values |x-3|<d where d can be computed.

but still by definition f(3)=0

the range of f(x) is [0,9)

Answer 2

This is a great question because it exposes the heart of calculus! Calculus is all about continuous functions, which is to say that they cover an infinite number of points within an interval, and that can be quite a confusing concept to work around.

To answer your question, consider the approach you might take to find that new absolute maximum (near the open critical point):

Let's say that a function f(x) is increasing until it reaches the critical point at f(5), before turning around and going back down. So, let's try finding the next biggest number by taking the value of f(4) (the critical point, but move 1 back on the x axis). Well, that can't be the maximum since there are all the values of x between 4 and 5 that should be larger. So, let's try f(4.9). Well, that can't be the maximum because you could still plug in f(4.99), f(4.999), f(4.9999), and you can start to see that we could continue doing that infinitely, getting slightly larger values of f every time.

This is why derivatives are extremely useful when working with continuous functions. Remember, the derivative gives the instantaneous rate of change at a point (how quickly the function is increasing or decreasing). If we took the derivative of each of those points (f(4), f(4.9), f(4.99), etc...) we would see that the derivatives keep getting smaller and smaller (the function is increasing more slowly, but still increasing). It is only once the derivative finally reaches 0 (or we hit an endpoint) that we can finally say for certain it is no longer increasing, and so there are no larger values.

So, in short, the maximum of a continuous function has to be a critical point or an endpoint, because otherwise you can always find a larger value of f(x) that is near the critical point.