Question

Decide whether the function f=sin(x)+cos(x) satisfies the hypotheses of the MVP on the interval [a,b]=[0,2π]


Then find all values of c in the interval [a,b] satisfying f′(c)=(f(b)−f(a))/b−a. If there is more more than one enter them as a comma separated list.
c=
Enter NONE if there are no such points in the interval.

Answer 1

@satellite73

Answer 2

of course is does
cosine and sine are both continuous and differentiable for all real numbers, and so certainly continuous aon \([0,2\pi]\)

Answer 3

you need a couple numbers here
\[ f(x)=\sin(x)+\cos(x)\] you need
\[f(2\pi)\] and
\[f(0)\] what do you get?

Answer 4

thats the derivative Cos[x] - Sin[x]

Answer 5

f(2pi) = 1
f(0)= 0

@satellite73

Answer 6

i don't think so

Answer 7

\[f(x)=\sin(x)+\cos(x)\]
\[f(2\pi)=\sin(2\pi)+\cos(2\pi)\]

Answer 8

i get \(1\) for that one

Answer 9

i used the initial equation

Answer 10

\[f(0)=\sin(0)+\cos(0)

Answer 11

\[f(0)=\sin(0)+\cos(0)\]

Answer 12

what did you get for that one?

Answer 13

lol its 1, my bad

Answer 14

k good

Answer 15

your final job is to take the derivative, set it equal to zero and solve

Answer 16

for 2pi i get 1 also

Answer 17

right

Answer 18

Cos[x] - Sin[x]

Answer 19

and \(1-1=0\) so set the derivative equal to zero

Answer 20

thats the derivative

Answer 21

yes

Answer 22

cos(x) - sin(x) = 0 now what

Answer 23

i would start with
\[\cos(x)=\sin(x)\] and then think of the two points on the unit circle where cosine and sine are the same

Answer 24

pi/4

Answer 25

right?

Answer 26

@satellite73