Question

# If $$f\left( x+y,\ x-y \right) =xy$$, then the arithmetic mean of $$f(x,y)$$ and $$f(y,x)$$ is

Solution

## Given: $$f\left( x+y,x-y \right) =xy$$Replacing $$x$$ by $$\dfrac { x+y }{ 2 }$$ and $$y$$ by $$\dfrac { x-y }{ 2 }$$Gives $$f\left( x,y \right) =\dfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 4 }$$Now the arithmetic mean is $$\dfrac { f\left( x,y \right) +f\left( y,x \right) }{ 2 } =\dfrac { \dfrac { { x }^{ 2 }-{ y }^{ 2 } }{ 4 } +\dfrac { { y }^{ 2 }-{ x }^{ 2 } }{ 4 } }{ 2 } =0$$Mathematics

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