Solveboth â ‹Q47 Let f be a one-one function such that f (x) f (y) + 2 = f - Maths -

Question

Solveboth
â€‹Q47 Let f be a one-one function such that
f (x) f (y) + 2 = f (x) + f (y) + (xy) ∀   x , y ∈
R ~ 0 and f (0) = 1, f(1) = 2 then prove that 3 ∫ f x dx  
- x   f x + 2 is constant

Q48 If e–ey f (xy) = ex f (x) +
ey f (y) ∀   x , y ∈ R , and f ' (1) =
e, determine f (x)

Neha Sethi · Accepted Answer

Dear student
48
Given e-xyf(xy)=e-xf(x)+e-yf(y)       
(1)Putting x=y=1 in (1), we get f(1)=0Now, f'(x)=limh→0f(x+h)-f(x)h=limh→0fx1+hx-f(x
1)h=limh→0ex+he-xf(x)+e-1-hxf1+hx-2xe-xf(x)+e-1f(1)h=limh→0ehf(x)+ex+h-1-hxf1+hx-f(x)-ex-1f(1)h=f(x)limh→0eh-1h+ex-1limh→0eh-hxf1+hxx×hx    As f(1)=0=f(x)
1+ex-1f'(1)x=f(x)+ex-1
ex   As f'(1)=eSo, f'(x)=f(x)+exx⇒e-xf'(x)=e-xf(x)=1x⇒ddxe-xf(x)=1xOn integrating, we havee-xf(x)=logx+C at x=1C=0So, f(x)=exlogx
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Solveboth ​Q47. Let f be a one-one function such that f (x) f (y) + 2 = f (x) + f (y) + (xy) ∀ x , y ∈ R ~ 0 and f (0) = 1, f(1) = 2 then prove that 3 ∫ f x dx - x f x + 2 is constant. Q48. If e–ey f (xy) = ex f (x) + ey f (y) ∀ x , y ∈ R , and f ' (1) = e, determine f (x).