Question

show that the function x^3-6x^2+12x+9 is increasing on all interval

Answer 1

To find out whether the function is increasing or decreasing, first you need to find the critical points. What are the critical points of this function?

Answer 2

i know that but the question is , to show that it is increasing at all intervals

Answer 3

how do i go about that?

Answer 4

Well after you have the critical points, you'll have a bunch of intervals from one critical point to the next. All you need to do is plug in any x-value within each interval into f'(x) and see what you get (because all the x-values within a given interval will follow the same trend in terms of slope).

If f'(x) > 0 (the slope is positive), then the interval is increasing.
If f'(x) < 0 (the slope is negative), then the interval is decreasing.

If f'(x) > 0 for every interval, then the function is always increasing.

Answer 5

ok thanks

Answer 6

Since you have to show that the function is increasing everywhere, you only need to establish that its derivative is positive.
\[f(x)=x^3-6x^2+12x+9~~\implies~~f'(x)=3x^2-12x+12=3(x-2)^2\]
What do you know about anything that gets squared?

Answer 7

waw. thats cool. all the values are positive

Answer 8

Well, everywhere except when \(x=6\).

Answer 9

@SithsAndGiggles
everywhere except x = 2
(just checking)

Answer 10

Right, I was thinking of a different question. Thanks