Suppose ' f ' is a function that satisfies the equation f ( x + y ) = f ( x ) + f ( y ) + x 2 y + x y 2 for all real numbers ' x ' and ' y '. If lim x → 0 x f ( x ) =

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Suppose ' f ' is a function that satisfies the equation f ( x + y ) = f ( x ) + f ( y ) + x 2 y + x y 2 for all real numbers ' x ' and ' y '. If lim x → 0 ​ x f ( x ) ​ = 1 , then

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Observe that f ( 0 ) = 0 and f ′ ( 0 ) = lim h → 0 ​ h f ( h ) ​ = 1 f ′ ( x ) = lim h → 0 ​ h f ( x + h ) − f ( x ) ​ ⇒ lim h → 0 ​ h f ( h ) + x 2 h + x h 2 ​ = x 2 + 1 Hence, f ( x ) = 3 x 3 ​ + x

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