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Question: If <img src="https://cdn.pureessence.tech/canvas_233.png?top_left_x=300&top_left_y=1375&width=300&he...

If then the arithmetic mean of f(x,y)f ( x , y ) and f(y,x)f ( y , x ) is.

A

xx

B
C

0

D

1

Answer

0

Explanation

Solution

Let x+y=u,x−y=vx + y = u , x - y = v

⇒ x=u+v2,y=u−v2x = \frac { u + v } { 2 } , y = \frac { u - v } { 2 },∴f(u,v)=(u+v2)⋅(u−v2)\therefore f ( u , v ) = \left( \frac { u + v } { 2 } \right) \cdot \left( \frac { u - v } { 2 } \right)

Now,f(x,y)+f(y,x)2=(x+y2⋅x−y2)+(y+x2⋅y−x2)2=0\frac { f ( x , y ) + f ( y , x ) } { 2 } = \frac { \left( \frac { x + y } { 2 } \cdot \frac { x - y } { 2 } \right) + \left( \frac { y + x } { 2 } \cdot \frac { y - x } { 2 } \right) } { 2 } = 0.