Solve this: â ‹Q50 Let f (xy) = xf (y) + y f (x) for all x, y∈R, - Maths -

Question

Solve this:

â€‹Q50 Let f (xy) = xf (y) + y f (x) for all x, y
∈ R, and f (x) be differentiable in (0 , ∞ ) then
determine f (x)

Lovina Kansal · Accepted Answer

Dear student
Given f(xy)=xf(y)+yf(x)Replacing x by 1 and y by x then we get xf(1)=0∴f(1)=0,x≠0   (∵x,y∈R+)Now, f'(x)=limh→0f(x+h)-f(x)h=limh→0fx1+hx-f(x)h=limh→0xf1+hx+1+hxf(x)-f(x)h=limh→0f1+hxhx+limh→0f(x)x=f'(1)+f(x)xSo, f'(x)=f'(1)+f(x)x⇒f'(1)=f'(x)-f(x)x⇒f'(1)x=xf'(x)-f(x)x2⇒f'(1)x=ddxf(x)xOn integrating w
r t x
, and taking limit 1 to x,we havef(x)x-f(1)1=f'(1)(logx-log1)⇒f(x)x-0=f'(1)logx    [As f(1)=0]Hence f(x)=f'(1)(xlogx)
Regards

Solve this: ​Q50. Let f (xy) = xf (y) + y f (x) for all x, y ∈ R, and f (x) be differentiable in (0 , ∞ ) then determine f (x).

Solve this:

Q50. Let f (xy) = xf (y) + y f (x) for all x, y $\in$ R, and f (x) be differentiable in (0 , $\infty$ ) then determine f (x).