Question

use the derivative of the function
y=f(x)to find the points at which f has a

local maximum
local minimum
point of inflection

y'=(x-1)^2(x-2)(x-4)

Answer 1

if f(x)=uvw
then
f'(x)=uvw'+ uwv'+ wvu'

Answer 2

I will assume you really meant to write the derivative y' and not the function y although I would have thought the book would ask you to find that derivative.

Wherever that derivative, y', is zero, the curve is horizontal
This is twice at x = 1 and at x =2 and at x = 4

Look at the second derivative to see what happens there.
y'=(x-1)^2 (x^2-6x+8)
so
y" = (x-1)^2(2x-6)+2(x-1)(x^2-6x+8)
now what is y"at x = 1?
y"(1) = Zero, so that is an inflection point
what is y" at x = 2?
y"(2) = -2+0 = -2
That is a maximum because y" is negative
what is y"(4)?
y"(4) = 9(2)+2*3(16-24+8)
y"(4) = 18+0 = 18
so it is Minimum at x = 4
y"