If f ( 2 x 2 + y 2 ) = 2 f { f ( x ) } 2 + { f ( y ) } 2 for all x , y ∈ R , find f ( x ) .

Question

If f ( 2 x 2 + y 2 ​ ) = 2 f { f ( x ) } 2 + { f ( y ) } 2 ​ for all x , y ∈ R , find f ( x ) .

TestHub · Accepted Answer

Differentiating both sides partially with respect to x , (keepiring y as constant), we get $f^{'} (\frac{x ^{2} + y ^{2}}{2}) \times \frac{1}{2} (2 x) = \frac{1}{2} [2 f (x) f^{'} (x) + 0] \Rightarrow x f^{'} (\frac{x ^{2} + y ^{2}}{2}) = f (x) f^{'} (x)$

Similarly differentiating both sides partially with respect to y (keeping x as constant) or interchanging $x$ and $y$ , we get $y f^{'} (\frac{x ^{2} + y ^{2}}{2}) = f (y) f^{'} (y) b$

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