Question

Find the extreme values of the function and determine the intervals where the function is increasing/decreasing expressed in interval notation.

x(x-2)^2

So far I've found y'=3x^2-8x+4.


Thanks

Answer 1

extreme where y' = 0
decreasing where y' < 0

y' = (3x-2)(x-2)

so, it should be clear where y'=0.
y' < 0 between the roots.

Easily verify your results using the graph at

http://www.wolframalpha.com/input/?i=x(x-2)%5E2

Answer 2

your derivative is correct. Where is it zero?

0 where x = 2 or x = 2/3

what is second derivative at those points?

y" = 6x - 8
at x = 2, that is POSITIVE so a minimum
at x = 2/3 that is NEATIVE so a maximum

at x ---> +oo that is +oo
at x ---> -oo that is -oo

Now sketch it :)