Question

Find the functions which satisfy the following conditions:

Answer 1

\[f(x+y) + f(x-y) = 2f(x)f(y) \]

And that: \[\lim_{x \rightarrow \infty} f(x) = 0 \]

Answer 2

f : R->R ???

Answer 3

It is cos(x)

Answer 4

@mukushla Yes.

Answer 5

And x,y are real.

Answer 6

let x=y=0\[2f(0)=2f^2(0)\]so f(0)=0 or 1

Answer 7

f(x) = 0 Would get us a constant function. That is one solution.

Answer 8

yeah let suppose f(0)=1

Answer 9

What about f(x) = cos(x)?

Answer 10

No.
Because, \[\lim_{x \rightarrow \infty} cosx \] is not 0.

Answer 11

It is zero

Answer 12

@mukushla How did you get that from x = 0?
@Zekarias No, it can be anything from zero to one.

Answer 13

oh my bad sorry :\

Answer 14

** @Zekarias -1 to 1 sorry.

Answer 15

setting x=y it gives\[f(2x)+1=2f^2(x)\]and y=-x gives\[f(2x)+1=2f(x)f(-x)\]so \[f(x)=f(-x)\]

Answer 16

is this correct ? and how it helps ... i dont know yet

Answer 17

how about we use the same technique as we used yesterday.

Answer 18

i think the only solution is f(x)=0

Answer 19

that's trivial ... let's hunt for some.