f-R-2-R-f-xy-xf-y-yf-x-f-x-

Question Number 805 by 123456 last updated on 17/Mar/15

f : R^{2} \to R

f (x y) = x f (y) + y f (x)

f (x) = ?

Answered by prakash jain last updated on 17/Mar/15

x = 1, y = 1

f (1) = 2 f (1) \Rightarrow f (1) = 0

x = - 1, y = - 1

f (1) = - 2 f (- 1) \Rightarrow f (- 1) = 0

f (0) = 0

y = - 1

f (x) = - f (x)

Divide by x y

\frac{f (x y)}{x y} = \frac{f (x)}{x} + \frac{f (y)}{y}

f (x) = {\begin{cases} x \ln ∣ x ∣ & x \neq 0 \\ 0 & x = 0 \end{cases}

f (x) = 0 \forall x \in R is also a solution .

Check

x y \ln ∣ x ∣∣ y ∣= x y \ln ∣ x ∣ + x y \ln ∣ y ∣

Commented by prakash jain last updated on 17/Mar/15

f (x) = k x \ln ∣ x ∣ is also a solutin where k constant .

Answered by prakash jain last updated on 17/Mar/15

Divide by x y f o r x y \neq 0

\frac{f (x y)}{x y} = \frac{f (x)}{x} + \frac{f (y)}{y}

g (x) = \frac{f (x)}{x}

g (x y) = g (x) + g (y)

\Rightarrow g (x) = k \ln x

\frac{f (x)}{x} = k \ln x

f (x) = k x \ln x