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Quadratic Polynomial

If 2f(x)=f(xy...

Question

If 2f(x)=f(xy)+f(x/y) for all positive values of x and y, f(1)=0 and f'(1)=1,find the value of f(e).

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

for all positive values of

x

y, f (1) = 0

f^{'} (1) = 1

f (e)

2 f (x) = f (x y) + f (\frac{x}{y}), x, y \in R^{+}

f (1) = 0

f^{'} (2) = 256,

f^{'} (256)

f (x) = cos (log x)

, then show that

f (\frac{1}{x}) f (\frac{1}{y}) - \frac{1}{2} (f (\frac{x}{y}) + f (x y)) = 0.

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

x, y \in R^{+}

f (1) = 0; f^{'} (1) = 2

f (e)

f (x) = cos (log x)

f (\frac{1}{x}) f (\frac{1}{y}) - \frac{1}{2} {f (\frac{x}{y}) + f (x y)} =

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