Addition formula for f(x+y) using mean-value theorem

[mathmathmad]

Homework Statement

suppose f and g are two differentiable functions with f(0)=0, g(0)=1 and f'(x)=g(x) and g'(x)=-f(x). For a fixed y in R put

F(x) = f(x+y) - f(x)f(y) - g(x)f(y)

Compute F'(x) and F''(x)
Then let

E(x) = [F(x)]^2 + [F'(x)]^2


Apply Mean Value Theorem to E and hence prove the addition formulae for f(x+y) and g(x+y)

Homework Equations

The Attempt at a Solution

I don't think what I've computed it correct, can anyone kindly please check?
http://img192.imageshack.us/img192/5237/rimg0036.jpg

 

[Coto]

First thing I notice is that you take a derivative with respect to y in F'(x). You are holding y fixed... that is treat it like a constant. It is equivalent to taking the partial derivative w.r.t. x.

Hope this gets you somewhere.

 

[mathmathmad]

that's what I thought but I was unsure about it
does that mean f(y) and g(y) will be treated as constants?
and is it correct that derivative of [F(x)]^2 is 2F(x).F'(x)?

 

[Coto]

Yes to all of the above :).

 

[mathmathmad]

I'm still stuck :( got loads of fs f's f''s with x y @_@ squares and all.. T_T

 

[Coto]

I'm still stuck :( got loads of fs f's f''s with x y @_@ squares and all.. T_T

Post your stuff. I'll take a look.

 

[mathmathmad]

http://img641.imageshack.us/img641/3946/rimg0001m.jpg
http://img291.imageshack.us/img291/5116/rimg0003a.jpg
it's messy though...

 

[Coto]

I would say everything looks great after the first page you posted.

Before getting started on the second page, have a look at the last line on your first page. You see the expression:

$$(F(c) + F''(c)) ?$$

Using the fact that f'(x) = g(x) and g'(x) = -f(x), and considering your formulas on lines 1 and 3 of the first page, you should be able to simplify this expression considerably.

Once you move on from here, you should consider taking a = 0 for your MVT and then use your information about f(0) and g(0) to further simplify the problem.

Unfortunately, I have to run! Good luck :).

 

[mathmathmad]

http://img52.imageshack.us/img52/5700/rimg0001.jpg
http://img188.imageshack.us/img188/5554/rimg0002q.jpg

still stuck :(

 

[Coto]

You should be able to find (provided you follow the steps given above) that E(b) = 0. Are you able to get this?

 

[mathmathmad]

yeah.. so the addtion formula of f(x+y) = f(x)g(y) + f(y)g(x) right? =)