Solve this: 1 Let fxy = fxfy V x,y∈R and f is differentiable at x=1 such that - Maths -

Question

Solve this:

1   Let   f xy   =   f x f y   V   x
, y ∈ R   and   f   is   differentiable
  at   x = 1   such   that   f 1   =
1   also   f 1 ≠ 0 , then   show   that
  is   differentiable   for   all   x ≠
0   Hence ,   determine   f x

Lovina Kansal · Accepted Answer

Dear student
Givne: f(xy)=f(x)f(y)We know, f'(x)=limh→0f(x+h)-f(x)h⇒f'(x)=limh→0fx1+hx-f(x
1)h⇒f'(x)=limh→0f(x)f1+hx-f(x)f(1)h   [using given relation]⇒f'(x)=limh→0f(x)f1+hx-f(1)h⇒f'(x)=limh→0f(x)x×f1+hx-f(1)hx⇒f'(x)=f(x)x f'(1)⇒f'(x)=f(x)x   ∵f'(1)=1 given⇒f'(x)f(x)=1xIntegrating both sides, we get∫f'(x)f(x)=∫1x⇒logf(x)=logx+logC⇒f(x)=xC   
(1)Putting x=1=y in the given relation we findf(1)=f(1)f(1)⇒f(1)=1Putting x=1 in (1), we getf(1)=C⇒C=1SO, (1) becomesf(x)=x

Regards

Solve this: 1 . Let f xy = f x f y V x , y ∈ R and f is differentiable at x = 1 such that f 1 = 1 also f 1 ≠ 0 , then show that is differentiable for all x ≠ 0 . Hence , determine f x .

Solve this:

$1 . Let f (xy) = f (x) f (y) V x, y \in R and f is differentiable at x = 1 such that f (1) = 1 also f (1 \neq 0,) then show that is differentiable for all x \neq 0 . Hence, determine f (x) .$