Question

(a) Find the critical numbers of the function f(x) = x^8(x − 3)^7. I dont have examples like this problem so Im having a bit of trouble, please help!

Answer 1

what is the interval?

Answer 2

is there an interval on which you are to find critical numbers, or there is no interval given >

Answer 3

anyway, I will tell you what to do, before you get to answer me.

Answer 4

the interval is not given

Answer 5

So, you have to:

1) Differentiate the function (take the derivative of the f(x) ).
2) Set f`(x)=0, and the x values that are solutions to this are critical numbers.
3) Critical numbers are also values where f`(x) is undefined - IFF f(x) is defined on that value(s) - but if f(x) is undefined at this value(s) then it is NOT a critical number.

Answer 6

Also, if you were given the interval [a,b] (specifically a closed interval) then a and b would automatically be (also) your critical numbers.

Answer 7

Is the outline/plan clear ?

Answer 8

yes! thank you!

Answer 9

ok, can you find f`(x) for me pliz ?

Answer 10

you can use product rule (or logarithmic differentiation)

Answer 11

working it out now

Answer 12

Product
d/dx [ f × g] = f` g + g` h

Answer 13

ok, take your time

Answer 14

i will post the provwe for product rule now.

Answer 15

so i have 8x^7 (x-2)^7 what do i do now?

Answer 16

y = f * g
(where f and g are some functions of x)
use logarithmic diff.
ln(y) = ln(f * g)
ln(y) = ln(f) + ln(g)

now, f and g are big function, so for any function H, there is a chain rule such that:
d/dx [ ln( H(x) ) ] = ( 1 / H(x) ) * H`(x)

So here we will use chain for f and g
differentiating:
(dy/dx) * y=f`/f + g`/g
dy/dx = (f`/f + g`/g) * y

then look at what y is from the beginning of the problem and substitute.
find common denom, and cancel things out

Answer 17

No, that is not correct. don't think so

Answer 18

f(x) = x^8(x − 3)^7
f`(x) = (x^8)` * (x − 3)^7 + x^8 * ( (x-3)^7 )`

Answer 19

f`(x) = 8x^7 * (x-3)^7 + x^8 * 7(x-3)^6

Answer 20

anyway, u got f`(x)
then find x values for
f`(x)=0
f`(x)=undefined

i got to go

Answer 21

ok, thank you!

Answer 22

I got x=0,3 and 24/15

Answer 23

(b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers?
At x=? the function has a local max, a local min or neither?