TargetIIT

if f'(x/y).f(y/x) = [im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7Bx%5E%7B2%7D+y%5E%7B2%7D%7D%7Bxy%7D[/im]     for all x,y belongs to positive R and f(1)= 1 then f[p]2[/p](x) is?????????

Does it look like [im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7B%28x%5E4+4%29e%5E%7B%5Cleft%28%5Ctan%5E%7B-1%7D%20%5Cfrac%7Bx%5E2%7D%7B2%7D%20-%20%5Ctan%5E%7B-1%7D%5Cfrac%7B1%7D%7B2%7D%5Cright%29%7D%7D%7B5%7D[/im]

dunno the ans....

[hide]i do not know wat u have done sir , but i think i can contradict your answer:



from given condition when u put  x=y   then u get f`(1)=1

but in your function we[b] do not[/b] get  f`(1)=1

so your answer must be wrong[/hide]

@ hassasin

f(1) = 1

f[p]'[/p](1)=2 .

putting x=1 in sir's solution we get 1.

Ok, I will outline the steps:

Put y=1, and obtain [im]http://codecogs.izyba.com/gif.latex?f%27%28x%29%20f%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%20%3D%20%5Cfrac%7Bx%5E2+1%7D%7Bx%7D[/im]...........1

Put x = 1 and obtain [im]http://codecogs.izyba.com/gif.latex?f%28x%29%20f%27%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%20%3D%20%5Cfrac%7Bx%5E2+1%7D%7Bx%7D[/im]...................2

Now let [im]http://codecogs.izyba.com/gif.latex?g%28x%29%20%3D%20f%28x%29%20f%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29[/im]...................3

So that [im]http://codecogs.izyba.com/gif.latex?g%27%28x%29%20%3D%20f%27%28x%29%20f%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%20-%20%5Cfrac%7B1%7D%7Bx%5E2%7D%20f%28x%29%20f%27%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%20%3D%20x%20-%20%5Cfrac%7B1%7D%7Bx%5E3%7D[/im]................4

Note by putting x =y that f'(1) =2

Now integrate between limits 1 and x to obtain

[im]http://codecogs.izyba.com/gif.latex?g%28x%29%20%3D%20f%28x%29%20f%20%5Cleft%28%5Cfrac%7B1%7D%7Bx%7D%20%5Cright%29%20%3D%20%5Cfrac%7Bx%5E2%7D%7B2%7D%20+%20%5Cfrac%7B2%7D%7Bx%5E2%7D%20-%20%5Cfrac%7B1%7D%7B2%7D[/im]...............5

From 1 and 5, we have

[im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7Bf%27%28x%29%7D%7Bf%28x%29%7D%20%3D%20%5Cfrac%7B2x%28x%5E2+1%29%7D%7Bx%5E4-x%5E2+4%7D[/im]

From here again by integrating between limits 1 and x, you will get f(x) and hence f[p]2[/p](x)