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Arithmetic Mean

If f x+y, x...

Question

If $f (x + y, x - y) = x y$ , then the arithmetic mean of $f (x, y)$ and $f (y, x)$ is

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Solution

Given: $f (x + y, x - y) = x y$

Replacing $x$ by $\frac{x + y}{2}$ and $y$ by $\frac{x - y}{2}$

Gives $f (x, y) = \frac{x^{2} - y^{2}}{4}$

Now the arithmetic mean is $\frac{f (x, y) + f (y, x)}{2} = \frac{\frac{x^{2} - y^{2}}{4} + \frac{y^{2} - x^{2}}{4}}{2} = 0$

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Similar questions

f (x + y, x - y) = x y

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

f (2 x + \frac{y}{8}, 2 x - \frac{y}{8}) = x y

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

\to F =^i F_{x} +^j F_{y} +^k F_{z}

is conservative, then

f (x)

be differentiable function such that

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

y

l t x \to 0 \frac{f (x)}{x} = \frac{1}{3}

f (1)

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

x, y \in R

{lim}_{h \to 0} \frac{f (h)}{h} = 3

, then the area bounded by the curves

y = f (x)

y = x^{2}

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