Question

Graph the function f(x)=x^3−2x and it's secant line through the points (-2,-4) and (2,4). Use the graph to estimate the x-coordinate of the points where the tangent line is parallel to the secant line.

Find the exact value of the numbers c that satisfy the conclusion of the mean value theorem for the interval [-2,2].

Answer 1

I need help finding 'c'

Answer 2

I know that the Mean Value Theorem says... \[\frac{ f(b)-f(a) }{ b-a }=f'(c)\]

Answer 3

just apply that...

Answer 4

But that gives me f'(c)

Answer 5

lemme check

Answer 6

here a=-2
b=2

Answer 7

Right, plugging all that in do you get f'(c) = 2 ?

Answer 8

i get 8

Answer 9

How?

Answer 10

f(b)=2^3-2(2)..rite?

Answer 11

f(a)=(-2)^3-2(-2)

Answer 12

Doesn't it already give us the coordinates for the points? (2,4) and (-2,-4)

Answer 13

Is it wrong to say \[\frac{ 4+4 }{ 2+2}=\frac{ 8 }{ 4 } = 2\] ? Or am I totally wrong right now? Lol

Answer 14

what did u do just now?

Answer 15

\[(x^3-2x)'|_{x=c}=\frac{f(2)-f(-2)}{2-(-2)} \\ (x^3-2x)'|_{x=c}=\frac{4-(-4)}{2-(-2)} \\ (x^3-2x)'|_{x=c}=\frac{4+4}{2+2}\]
that is right for the right hand side

Answer 16

Well I plugged in f(a) = -4, f(b) = 4, and a=-2, b=2 in the mean value theorem formula

Answer 17

what happens to the left hand side?

Answer 18

do i just derive it?

Answer 19

you still have to differentiate the left hand side and plug in c afterwards as the equation above suggests

Answer 20

3x^2 - 2 right?

Answer 21

yep and plug in c (though that part doesn't matter to much; it is just what they call it in the formula above)
yep now solve 3x^2-2=2

Answer 22

or you named it c
3c^2-2=2

Answer 23

Ohh so then its +/- 2/sqrt3 right?

Answer 24

are both of those in the given interval?

Answer 25

hmmm... yes?

Answer 26

ok then cool stuff

Answer 27

Thats it? :O

Answer 28

that is it

Answer 29

though there are two problems

Answer 30

did you do the first problem?

Answer 31

Yay!!! Thats correct! :D Thank you so much!

Answer 32

we did the second problem
we found the exact solutions

Answer 33

Yeah Im always better in graphing haha

Answer 34

alright

Answer 35

Thanks again for the help! :)

Answer 36

np

Answer 37

tracy?got the answer for the second part?

Answer 38

Yeah! I got it, its + or - 2/sqrt3

Answer 39

Thank you too @chethus!