Find where f is increasing, it's minima...

I have such function f(x) = x^6(x-4)^3 for the interval [-11, 10],

I need to evaluate on which intervals the function is increasing, on which one is positive and where is the minimum.

I tied to find the derivative , so f'(x) = 6x^5(x-4)^3*3(x-4)^2*x = and wrote it as 18x^6(x-4)^3(x-4)^2

even before calculating the critical points it seems that I complicate something,

Could you help me by indicating one simpler way of factoring and differentiating this?

Re: Find where f is increasing, it's minima...

Hi Dokrbb

If I did understand the question right,

We are looking for the intervals where the function f(x) given is increassing.

So as you (should) know (or remember now that I say it : ) f(x) is increasing in an interval if its derivative is Positive (so if f'(x) > 0) So I had try to find the value for f'(x) =0 then look beetween the value of x for f'(x) =0 and see if its positive.

That should get you started.

Hope I helped

Re: Find where f is increasing, it's minima...

well, your advice was helpful and not helpful in the same time:

it's helpful since from what you said I am on the right track, but it's not really,( :) I'm joking, sorry) because I already tried to obtain those points and I obtained the critical points as for

x=-1, x=0 and x=4, but it's something wrong with these points, that's why I posted the derivative I obtained (I suppose that's where I got wrong...) and would really appreciate your help in getting those critical points,

Re: Find where f is increasing, it's minima...

I had recheck the derivative you found!

remeber the (u*v)' = u'v+uv'

Re: Find where f is increasing, it's minima...

There is some thing wrong with your critical points

The derivative f'(x) = 6x^5 (x-4)^3 + 3 x^6 (x-4)^2

For critical points f'(x) = 0 gives

3x^5 (x-4)^2 [ 2 ( x-4 ) + x ] = 0 That is 3x^5 (x-4)^2 ( 3x-8) = 0 and that gives x = 0, 4 and 8/3 = 2.67

Now pick up any convenient in the intervals and evaluate f'(x) the interval where it is > 0 the function is increasing and where it is < 0 the function is decreasing.

Re: Find where f is increasing, it's minima...

Thank you guys, now I see where I was wrong...