Question

Verify that the Intermediate Value theorem applies to the indicated interval and find the value of c guaranteed by the theorem.

f(x) = x^2 - 6x + 8, [0,3], f(c) = 0

Answer 1

@Neemo @jim_thompson5910

Answer 2

@Hero

Answer 3

what is your indicated interval?

Answer 4

it is 0,3

Answer 5

i then found that it takes all values between -1 and 8. and 0 iis between these values... so what wwould i put as my answer?

Answer 6

f(0) = 8
f(3) = -1
f(2) = 0

f(2) belongs to [f(0), f(3)]

Answer 7

Ya

Answer 8

is that all I put for my answer

Answer 9

Basically because f(0) = 8 and f(3) = -1, this means that f(c) = 0 must lie between the two somewhere by the intermediate value theorem

Answer 10

ok so thats basically my answer then, thats it.

Answer 11

So you can either factor, use a graph, or the quadratic formula to find that f(2) = 0, so c = 2

Answer 12

ok thx. i have one more, its easier. can you help with this? :

Answer 13

sure

Answer 14

Describe the difference between a discontinuity that is removable and one that is a nonremovable.? In your explanation, give examples of the following: a) a function with a nonremovable discontinuity at x=2. b) a function with a removable discontinuity at x= -2. c) a function that has both of the characteristics described in a. and b.

Answer 15

x/(x-2) has a non-removable discontinuity at x = 2 since plugging in x = 2 results in a division by zero error (and you can't simplify to make the error go away)

Answer 16

something like (3x+6)/(x+2) has a removable discontinuity at x = -2 since

(3x+6)/(x+2)

(3(x+2))/(x+2)

3

So (3x+6)/(x+2) = 3, but we must restrict that \(x \neq -2\) to make sure that the two domains match up

Answer 17

ok

Answer 18

a function that has both x = 2 as a non-removable discontinuity and x = -2 as a removable discontinuity is something like


(5x+10)/(x^2-4)

since

(5x+10)/(x^2-4)

(5(x+2))/((x+2)(x-2))

5/(x-2)

So (5x+10)/(x^2-4) = 5/(x-2), but \(x \neq 2\) and \(x \neq -2\) to avoid division by zero errors

Answer 19

thx so much

Answer 20

np