Question

What are the values and types of the critical points, if any, of `f(x) = (x^4/5)(x-4)^2`?

Answer 1

Turning points at `x = 8/3 and x = 4`.
Flat, at x = Look at the two graphs.

Explanation:

It is easy to show that

`f' = 0, x = 0, 8/3 and 4`.

Further, f'' is 0 only at x = 0. Yet, f''' = 0 here but

the next ( even order ) f'''' is not.

So, origin is not a point of inflexion, and so, at this turning point,

the curve is flat, in the `epsilon-`neighborhood `(-epsilon, epsilon)`

This flatness is zoomed in the second graph.

graph{5y-x^4(x-4)^2=0 [-40, 40, -20, 20]}

graph{5y-x^4(x-4)^2=0 [-5, 5, -2.5, 2.5]}