f-R-R-g-R-R-f-xy-f-x-g-x-f-y-g-y-g-xy-f-x-g-y-f-y-g-x-d-fg-dx-

Question Number 818 by 123456 last updated on 17/Mar/15

f : R \to R

g : R \to R

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

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

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

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

x = 1, y = 1

f (1) = 2 f (1) g (1)

g (1) = 2 f (1) g (1)

f (1) = g (1)

f (1) = 2 {[f (1)]}^{2}

f (1) = 0 \lor f (1) = \frac{1}{2}

If f (1) = 0, g (1) = f (1) = 0, f (x) = g (x) = 0

If f (1) = \frac{1}{2}, g (1) = f (1) = \frac{1}{2}

g (x) = \frac{1}{2} [f (x) + g (x)] \Rightarrow f (x) = g (x)

f (x) = {[f (x)]}^{2} + \frac{1}{4}

f (x) = g (x) = \frac{1}{2}

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

Two solutions

f (x) = g (x) = 0

f (x) = g (x) = \frac{1}{2}

\frac{d (f g)}{d x} = 0