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
Best Answer
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 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.