Question

Find the critical numbers of the function and describe the behavior of f(x) = x8(x - 3)7 at these numbers.

Answer 1

\[f(x) = x^8(x - 3)^7\]\[f'(x)=8x^7(x-3)^7+7x^8(x-3)^6\] via hte product rule

Answer 2

factor the derivative, so you can find the zeros easily

Answer 3

And that's the first derivative. What is it I have to do to find the zeros? That has always confused me

Answer 4

you have to write it in factored form first

Answer 5

it is pretty clear just from looking at it that one zero is \(0\) and also one zero is \(3\) but you have to factor it
both terms have a common factor of \(x^7(x-3)^6\)
factor that out

Answer 6

it is algebra at this point, i can show you the method if you like

Answer 7

I'm not good at algebra, my strong points are in geometry, if you would please I would like that

Answer 8

ok

Answer 9

\[f'(x)=8x^7(x-3)^7+7x^8(x-3)^6\]
common factor in both terms is \(x^7(x-3)^6\)
so first step is
\[8x^7(x-3)^7+7x^8(x-3)^6=x^7(x-3)^6\left(\text{something}\right)\]

Answer 10

ok to this step?

Answer 11

Yes

Answer 12

ok good and the "something" which you get by thinking
has to be \(8(x-3)+7x\)

Answer 13

Oh okay. That makes sense now.

Answer 14

\[8x^7(x-3)^7+7x^8(x-3)^6=x^7(x-3)^6\left(8(x-3)+7x\right)\]then algebra in the second part, gives
\[x^7(x-3)^6(8x-24+7x)=x^7(x-3)^6(15x-24)\] and now the other zero is easy to find

Answer 15

And they would be... -3 and -24?

Answer 16

oh no

Answer 17

Calculus really makes me feel stupid.

Answer 18

set each factor equal to zero and solve:
\[x^7=0\iff x=0\\
x-3=0\iff x=3\\
15x-24=0\iff 15x=24\iff x=\frac{24}{15}=\frac{8}{5}\]

Answer 19

that is just because you have to use all that elementary algebra you may have forgotten
that is why they taught that first

Answer 20

the ideas of calc are not that hard, the mechanics are the confusing part

Answer 21

Yeah, the logic doesn't seem logical to me. So then are those the critical points?

Answer 22

yes

Answer 23

Okay. And it's increasing until 0, decreasing until 8/5, and increasing until three?

Answer 24

yes, i believe it is, although it is increasing after 3 as well, the derivative does not change sign at 3
you can check the picture here
http://www.wolframalpha.com/input/?i=x^8%28x+-+3%29^7+domain+-1..4

Answer 25

Okay, thank you. That helped me a lot

Answer 26

@satellite73 please close the question in biology about taylor swift
thank you and have a nice day