Question

Verify that the function g(x)= 3x/x+7 satisfies the hypothesis of the Mean Value Theorem on the interval [-1,2]. Then find all the numbers c that satisfy the conclusion of MVT.

Answer 1

Do you know what the Mean value theorem states?

Answer 2

yeah but idk how to apply

Answer 3

is this function continuous at every point of the closed interval [-1,2]?

Answer 4

everywhere except x=-7

Answer 5

there's a hole there

Answer 6

oh so it is continuous on the interval

Answer 7

So it is continuous on the interval [-1,2]. Which means it is differentiable on (-1,2). So find the derivavtive of the function.

Answer 8

derivative*

Answer 9

would the derivative just be 3?

Answer 10

Is this your function?
\[\frac{ 3x }{ x+7 }\]

Answer 11

yes

Answer 12

Have you learned the quotient rule yet?

Answer 13

yeah

Answer 14

ooooh oops

Answer 15

I suggest you do it by hand to practice, but if you are already proficient in calculating the derivative, this could save time. Plus it shows the steps.
http://www.derivative-calculator.net/

Answer 16

\[\frac{ 21 }{ x^2+14x+49 }\]

Answer 17

i tried it by hand

Answer 18

Plug your function into the website i linked above, it will show you all the steps.

Answer 19

i got it right

Answer 20

they just didn't factor the denomiator

Answer 21

so what do I do next

Answer 22

So now that you have that. Find:
\[\frac{ f(b)- f(a) }{ b-a }\]

Answer 23

what was the point in taking the derivative of the original function then?

Answer 24

Because after you find the value of
\[\frac{ f(b)-f(a) }{ b-a }\]
you will set that equal to the derivative and solve for x.

Answer 25

also i got 15/6

Answer 26

so then I set 15/6 = to the derivative?

Answer 27

yep

Answer 28

ahh okay I'm confused on how to solve here

Answer 29

for x

Answer 30

you could multiply both sides by 1/21 to get rid of the 21 in the numerator.

Answer 31

okay so then I have (x+7)^2=15/126

Answer 32

I unfactored it bc it seemed easier

Answer 33

oops i messed up

Answer 34

it would = 315/6

Answer 35

then what should I do?

Answer 36

move the fraction other side so your function is equal to zero.

Answer 37

okay then what

Answer 38

use the quadratic equation to find x

Answer 39

wait so i would be using x^2+14x-3.5 for the quadratic formula?

Answer 40

i just want to make sure before I go at it

Answer 41

or should I stick with fractions?

Answer 42

Alright so this whole time i have just been trusting that you have done all of the steps correctly without actually doing all of the work.

Answer 43

i just did:
\[\frac{ f(b)-f(a) }{ b-a }\]
and i got 1/9

Answer 44

wait how

Answer 45

do you know what f(b) - f(a) means?

Answer 46

yeah you plug in the points from the given interval

Answer 47

\[\frac{ 3(2) }{ (2) +7 } -\frac{ 3(-1) }{ (-1)+7 } = \frac{ 6 }{ 9 } - \frac{ -3 }{ 6 } = \frac{ 1 }{ 3 }\]
and 2 -(-1) = 3 therefore (1/3)/3 = (1/9)

Answer 48

isn't it 2+1 in the denominator

Answer 49

yes, which is 3 just like i posted above.

Answer 50

\[x=-7\pm \sqrt{26}\]

Answer 51

??

Answer 52

@royalranger