Question

Let f(x) = (1 − x2)2/3. Which type of non-differentiable point exists at x = 1?.

Answer 1

`f(x)=(1-x^2)^{(2)/(3)}\implies \cfrac{df}{dx}=\cfrac{2}{3}(1-x^2)^{-(1)/(3)}\implies \cfrac{df}{dx}=\cfrac{2}{3\sqrt[3]{1-x^2}}`

when it comes to a rational expression, we can get critical points from, zeroing the derivative "and" from zeroing the denominator alone, however the denominator provides critical valid points that are either "asymptotic" or "cuspics", namely that the function is not differentiable or not a "smooth line" at such spot.

if we get the critical points from the denominator on this one, we get x = ±1, both of which are cuspics. Check the picture below.