Question

Find all critical points of f (x) = x^4 - 4 x^3 + 4x^2

Answer 1

\[x ^{4} -4x^{3} +4x^{2}\]

Answer 2

using calculus?

Answer 3

Yes, may be .. but I am not sure

Answer 4

How can we find critical points of a function?

Answer 5

use first derivitive test, set the derivitive = 0 to find all possible critical points, then test values

Answer 6

okay, thanks .. it means I need to first understand the concept of derivative and limits

Answer 7

\[f \prime(x) = 4x ^{3}-12x ^{2}+8x\]
\[=4x(x-1)(x-2)\]
x=0, x=1, x=2
sign chart to find that f'(x)<0 on (-inf,0), f'(x)>0 on (0,1), f'(x)<0 on (1,2), and f'(x)>0 on (2,inf)
therefore, by the first derivitive test there are critical points at x=0, x=1, and x=2

Answer 8

I should learn derivative and limits. I am not understanding it. Thank you for your help and time :)

Answer 9

standard derivative is simply (n-1)*x^(n-1) do that for the polynomial and then set it to 0 and thats it :)