First slide

Functions (XII)

Question

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

Moderate

A

y
B

x
C

0
D

None of these

Solution

Let x + y = a and x – y = b
$\Rightarrow x = \frac{a + b}{2} and y = \frac{a - b}{2}$
f (x + y, x – y) = xy
$\begin{array}{l} \Rightarrow f (a, b) = \frac{a + b}{2} \cdot \frac{a - b}{2} = \frac{a^{2} - b^{2}}{4} \\ ∴ f (x, y) = \frac{x^{2} - y^{2}}{4} and f (y, x) = \frac{y^{2} - x^{2}}{4} \end{array}$
Arithmetic mean of f (x, y) and f (y, x) is zero.

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