Question

Show whether Rolle's theorem can be applied to f(x). If it can, determine any values of c in the interval [0,2pi] for which f'(c)= 0.
f(x)= cosx

Answer 1

so for Rolle's th. to be applicable, f must be continuous and differentiable. I believe this function is. What else needs to be proven?

Answer 2

I could use some help. @turtlescarf, do you have any ideas?

Answer 3

c=0 and 2pi

Answer 4

cosx is differentiable and continuous. f'(c)=sinc. then set sinc=0 and solve for c. you get 0 and 2pi :)

Answer 5

oh and f(0) must equal f(2pi)

Answer 6

sorry let me do the problem again, i was using medium value theorem

Answer 7

okay, cos0 and cos2pi both equal 1. therefore, f'(c)=0

Answer 8

oh thank you for the responses! I haven't read them yet so give me a second.

Answer 9

so I just need to prove that cos(0)= cos(2pi)? along with the fact that f is continuous and differentiable. And that would prove Rolle's Th. Applicable?

Answer 10

oh but then what would c be? how do I solve for that?

Answer 11

find the derivative and set = to 0 and solve for x?

Answer 12

@turtlescarf ?

Answer 13

set f'(c)=0 so you can solve for c

Answer 14

you're right, but you solve for c instead of x

Answer 15

so if f'(x)= -sinx and I set equal to 0, how do I solve with the 'sin'
wait how do I solve for the c?

Answer 16

f'(c)=-sinc

Answer 17

you replace the x with c

Answer 18

then, using the theorem, you set -sinc=0

Answer 19

and then set to 0?

Answer 20

divide both sides by -1 and you get sinc=0

Answer 21

so c is 0 and 2pi

Answer 22

yes.

Answer 23

oh ok so c could be either because they are both in the interval?

Answer 24

0 and 2pi are technically the same thing on the circle*

Answer 25

hm i'd say c is both, even though both values are the same, just in case your teacher/whoever prefers a certain answer

Answer 26

basically, yes.

Answer 27

oh ok. Thank you so much for your help! I'll give you a medal and then close this question because I have one more problem to get to tonight!

Answer 28

in response to "so I just need to prove that cos(0)= cos(2pi)? along with the fact that f is continuous and differentiable. And that would prove Rolle's Th. Applicable?" Yes. and also show that f(a)=f(b), which in this case is f(0)-f(2pi).

Answer 29

you're welcome