Question

PLease help If f(x) = |(x2 − 9)(x2 + 1)|, how many numbers in the interval [–1, 1] satisfy the conclusion of the Mean Value Theorem?

Answer 1

I understand mean value theorem but I dont know where to start

Answer 2

You understand mean value theorem?
Ok great.

So what slope value does the `secant line` that connects the ends points give us?

Answer 3

\(\large\rm \frac{f(1)-f(-1)}{1-(-1)}=?\)

Answer 4

solving that would give me f'(c)

Answer 5

Im just confused how absolute value effects the entire thing

Answer 6

Yes, but give me the number first for the slope,
we'll deal with derivative later.

Answer 7

(|-8/2|-0)/(2)

Answer 8

Woops, shouldn't get 0 in the numerator.
I think you forgot to square your -1, yes?

Answer 9

i did haha

Answer 10

(|-16|-|-16|)/2

Answer 11

Ok good good good.
So we're getting \(\large\rm \frac{f(1)-f(-1)}{1-(-1)}=0\)

Answer 12

So notice, this problem boils down to a special case of the Mean Value Theorem, known as Rolle's Theorem.

We're looking for zero slopes in this interval,
better known as `critical points`.

Answer 13

So now we at least know what we're looking for :)
Derivative is going to be a little bit of a pain.

Let me remind you of absolute value derivative.

Answer 14

\[\large\rm \frac{d}{dx}|x|=\frac{x}{|x|}\]If our inner function is more than just x, don't forget to chain rule,\[\large\rm \frac{d}{dx}|stuff|=\frac{stuff}{|stuff|}\cdot(stuff)'\]

Answer 15

so i have to get f"(x)?

Answer 16

So it's going to look pretty scary, should simply down nicely though.\[\large\rm \frac{d}{dx}|(x^2-9)(x^2+1)|=\frac{(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot[(x^2-9)(x^2+1)]'\]

Answer 17

No... why would you need f''?

Answer 18

We're looking for critical points :o
We don't care about inflection or anything like that.

Answer 19

OKay sorry

Answer 20

I know it looks like a big mess,
but hopefully you can at least see how it follows the rule that I posted for absolute derivative.

Answer 21

So now we have to multiply by the derivative of the inner function,\[\rm \frac{d}{dx}|(x^2-9)(x^2+1)|=\frac{(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot\color{royalblue}{[(x^2-9)(x^2+1)]'}\]We'll have to product rule here.

Answer 22

solving it now

Answer 23

(4 x (-9+x^2) (-4+x^2) (1+x^2))/(abs((-9+x^2) (1+x^2)))

Answer 24

looks even messier lol

Answer 25

I can't quite figure out what you did :( hmm

Answer 26

i got the derivative 4x(x^2-4)

Answer 27

I then multiplied it by the main function

Answer 28

\[\rm =\frac{(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot\color{royalblue}{[(x^2-9)(x^2+1)]'}\]Product rule:\[\rm =\frac{(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot\color{royalblue}{[(x^2-9)'(x^2+1)+(x^2-9)(x^2+1)']}\]That's the setup yes?

Answer 29

Oh oh oh, don't multiply stuff together :( yikes

Answer 30

yes but i thought you got the derivative and then multiplied

Answer 31

Finishing up product rule gives us,\[\rm =\frac{(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot\color{royalblue}{[(2x)(x^2+1)+(x^2-9)(2x)]}\]Factor a (2x) out of each term,\[\large\rm =\frac{(2x)(x^2-9)(x^2+1)}{|(x^2-9)(x^2+1)|}\cdot\left[x^2+1+x^2-9\right]\]Oh oh oh nevermind :) I see where your 4 is coming from.
Ok ok good good good

Answer 32

\[\large\rm =\frac{(4x)(x^2-9)(x^2+1)(x^2-4)}{|(x^2-9)(x^2+1)|}\]

Answer 33

This is our derivative function.
We're looking for zero-slope,\[\large\rm 0=\frac{(4x)(x^2-9)(x^2+1)(x^2-4)}{|(x^2-9)(x^2+1)|}\]Which leads to the numerator equaling zero,\[\large\rm 0=4x(x^2-9)(x^2+1)(x^2-4)\]

Answer 34

Solve for x,
and then determine which of those critical points you're allowed to use.
(Only the ones within the interval [-1,1].)

Answer 35

The derivative says "how many",
they didn't actually ask you to find them,
which leads me to believe that there is probably an easier way to do this problem XD lol

Answer 36

the directions say*

Answer 37

haha. The work was killer but Im glad we did it that way

Answer 38

is it only 1 root then? x=0?

Answer 39

yay good job \c:/

x=3,-3 and x=2,-2 and x=0
only 0 lies in the interval

Answer 40

Youre the best. Thanks!

Answer 41

not to butt in but i think you are doing way way too much work here

Answer 42

:O. Can you tell me the shorter way haha

Answer 43

this is your function right?\[f(x)= |(x^2 − 9)(x^2 + 1)|\]

Answer 44

yes

Answer 45

and the interval is \([-1,1]\) right?

Answer 46

yes

Answer 47

on that interval, \(x^2-9\) is negative
so \(|x^2-9|=9-x^2\) as far as you are concerned

Answer 48

and of course \(x^2+1\geq 0\) for all \(x\) so your function is just \[(9-x^2)(x^2+1)\] which is a lot easier to deal with

Answer 49

True, heh

Answer 50

haha wish i knew that. Thanks for pointing it out