Solve this: Q 52 If 2fx=fxy+fxy for all x, y ∈R+, f (1) = 0 and f' (1) = - Maths -

Question

Solve this:
Q 52 If 2 f x = f x y + f x y for all x, y ∈ R + , f (1) = 0
and f' (1) = 1, then find f (e) and f' (2)

Neha Sethi · Accepted Answer

Dear student
We have, fxy=2fx-fxy 
1Putting x=1 in 1, we getfy=2f1-f1yfy=-f1y    Given: f1=0 
2In 1 we exchange x and y to getfxy=2fy-fyx⇒fxy=2fy+fxy   using 2 
3Now we subtract 1 from 30=2fy+fxy-2fx+fxy⇒0=2fy-2fx+2fxy⇒fx-fy=fxy   
4We have,f'x=limh→0fx+h-fxh=limh→0fx+hxh     using 4=limh→0f1+hxx×hx=f'1xi
e
 f'x=1x    as f'1=1Now on integrating we getf(x)=lnx+cUsing the condition f1=0, gives f1=ln1+c0=c   as ln1=0So, f(x)=lnx⇒fe=lne=1   as lne=1and f'x=1xSo, f'2=12
Regards

Solve this: Q.52. If 2 f x = f x y + f x y for all x, y ∈ R + , f (1) = 0 and f' (1) = 1, then find f (e) and f' (2).

Solve this:
Q.52. If $2 f (x) = f (x y) + f (\frac{x}{y})$ for all x, y $\in R^{+}$ , f (1) = 0 and f' (1) = 1, then find f (e) and f' (2).