How to tell if a function has a cusp without a graph?

Question

For my calculus exam, I need to be able to identify if a function is indifferentiable at any point without a graph. I thought this would be rather simple, but I messed up on the question x^(2/3) because I did not realize it had a "cusp" at x = 0.

How would I identify, or look for cusps based on the formula of a function, without graphing it?

I plugged in other similar formulas into a graphing calculator, like x^(1/3), x^(1/4), x^(3/4) but couldn't identify a simple pattern I could quickly use on my test

Answer 1

I need to be able to identify if a function is indifferentiable at any point

The common way to do that is to actually determine the derivative and inspect it for singularities. This is generally easy with elementary functions.

In your example:

$$ f(x) = x ^ \frac{2}{3} $$ $$ f'(x) = \frac{2}{3} x ^ \frac{-1}{3} = \frac{2}{3 \sqrt[3] x} \;\; for \;\; x \ne 0 $$

On cursory inspection (or by applying the definition), it's obvious that $f'(0)$ doesn't exist, so $f(x)$ is not differentiable at $0$.

How would I identify, or look for cusps based on the formula of a function, without graphing it?

How would you if you could graph it?

You can't draw an infinite graph at infinite resolution. Maybe the "cusp" is at $x = 10^{10^{10}}$, or maybe it's for $f(x) = 0.00001$.

Or, try graphing $f(x) = x \sin \frac{1}{x}$ and finding the "cusp" there.

Answer 2

Probably your problem is that you are not checking the conditions for theorems to hold, and that is why you do not see where they fail.

$\frac{d(x^n)}{dx} = n x^{n-1}$ for every positive integer $n$.

$\frac{d(x^n)}{dx} = n x^{n-1}$ for every integer $n$ and variable $x \ne 0$.

$\frac{d(x^n)}{dx} = n x^{n-1}$ for every rational $n$ and real variable $x \ne 0$ such that $x^n$ is defined.

$\frac{d(x^n)}{dx} = n x^{n-1}$ for every real $n$ and real variable $x > 0$.

Clearly if you apply the appropriate theorem for $n = \frac23$, you will see that $( x \mapsto x^{2/3} )$ is guaranteed to have a derivative everywhere except $x = 0$. Thus you can focus on that single point and check whether it has a derivative there or not. In this case it does not, but in general it might or might not; you've to check.