Question

b) Find the value(s) of c that satisfy the conclusion of the MVT for the function
f(x) =sqrt(x-1) on the interval [1; 3].
c) How many real roots does the equation x^5 + 2x + 3 = 0 have?

Answer 1

first, do you know what the mean value theorem says?

Answer 2

basically an equation will have a tangent line equal to
the average slope of the line

Answer 3

so first lets find the slope of the line in equation

Answer 4

so (0-sqrt2)/1-3

Answer 5

yup so that is \[\sqrt{2}/2\]

Answer 6

yeah

Answer 7

now lets take the derivative of your function

Answer 8

(.5)(x-1)^(-.5)

Answer 9

so 1/sqrtx-1

Answer 10

1/2(ssqrtx-1)

Answer 11

great, so now solve for x when the derivative equals sqrt(2)/2

Answer 12

so plug sqrt2/2 into f'(x)?

Answer 13

f'(sqrt/2)??

Answer 14

not exactly, you are looking for what x value will guve you a derivative of sqrt(2)/2

Answer 15

so solve\[\sqrt{2}/2=1/(2\sqrt{x-1})\]

Answer 16

ok, then what? while i have your attention, haha

Answer 17

then you have your answer

Answer 18

oh i see, so my x value once solves for is my value "c"

Answer 19

then what about the real roots equation, i feel like its just 5, but i don't really know how to prove it

Answer 20

first, tell me what math course you are in because there are a few different ways to go about this problem depending on what course this is in

Answer 21

calc 1

Answer 22

ok. lets think about this, what happens to f(x) as x approaches infinity?

Answer 23

it goes toward infinity

Answer 24

and what about as f(x) approaches negative infinity

Answer 25

negative infinity

Answer 26

ok, good, do you know what an inflection point is?

Answer 27

f''

Answer 28

f''(x)=0

Answer 29

yeah so that is when the derivative is going to change sign (become positive or negative)

Answer 30

can you tell me what the first derivative of this function is?

Answer 31

5x^4 + 2

Answer 32

i have to leave for class in 5 minutes

Answer 33

does that equal zero for any real x value?
(we are almost done)

Answer 34

yeah i think it does

Answer 35

since the derivative is always positive here, we know that there arent any dips in the function so it can only cross that x axis once. This means that there is only one real root

Answer 36

x^4 is always postive so 5x^4 +2 is always positive

Answer 37

ohhhhhh

Answer 38

wow

Answer 39

if you look at this http://www.wolframalpha.com/input/?i=x%5E5%2B2x%2B3 you will see that it only crosses the x axis once

Answer 40

so we can use the derivative to see that the function is always increasing

Answer 41

right right, i understand so basically it was asking how many times does it cross the x axis, that makes so much more sense

Answer 42

i get where u are going with the +/- infinity too now, ahha

Answer 43

if it is always increasing from negative infinity to infinity, it will only cross the x axis once

Answer 44

Thank you very much

Answer 45

no prob, enjoy class