What are the relative and absolute extrema's for these functions?

Question

I need to know when the function is increasing/decreasing and when relative maxima and minima occur, without sketching the graph. I know you do this by finding the derivative and setting it to equal zero, however I am not getting proper x values. Here are the fuctions I need help with:

Relative Extrema:
1) y = x^3-6x^2 + 12x - 6

2) y= 3x/2x+5

3) y= xlnx

Absolue Extrema:
4) f(x) = x ^4/3, [-8,8]

Thank you.

Answer 1

1) y = x^3 - 6x^2 + 12x - 6

To determine intervals of increase and decrease, as well as local extrema, we take the first derivative and then make it 0.

y' = 3x^2 - 12x + 12

Making y' = 0, we have

0 = 3x^2 - 12x + 12

Dividing both sides by 3,

0 = x^2 - 4x + 4
0 = (x - 2)^2, therefore
x = 2.

Our critical point is x = 2. That means we test a value less than 2 and a value greater than 2, to determine intervals of increase/decrease. If the value is positive, it is increasing on that interval; if negative, it is decreasing.

Test x = 0; then, for y' = 3x^2 - 12x + 12, we get the value
0 - 0 + 12 = 12, which is positive. Therefore our function is increasing on (-infinity, 2].

Test x = 10000000. We're going to get a really big positive number, so our function is increasing on [2, infinity).

That means our function is always increasing, and we have no relative extrema.