If $ f(x+y)=f(x)+f(y)+|x| y+x^{2} y^{2}, \forall x, y \in R $ and $ f^{\prime}(0)=0 $, then

Question

If $ f(x+y)=f(x)+f(y)+|x| y+x^{2} y^{2}, \forall x, y \in R $ and $ f^{\prime}(0)=0 $, then, not be differentiable at every non-zero $ x $., differentiable for all $ x \in R $., twice differentiable at $ x=0 $., none of the above.

MARKS App · Accepted Answer

Given: f x + y = f ( x ) + f ( y ) + x y + x 2 y 2 and f ' ( 0 ) = 0 Substitute, x = y = 0 f ( 0 + 0 ) = f ( 0 ) + f ( 0 ) + 0 + 0 ⇒ f ( 0 ) = 0 f ' ( x ) = lim h → 0 f x + h - f ( x ) h f ' ( x ) = lim h → 0 f ( x ) + f ( h ) + x h + x 2 h 2 - f ( x ) h = lim h → 0 f ( h ) h + x + x 2 h = f ' ( 0 ) + x f ' ( x ) = 0 + x = x f ' ( x ) = - x x 0 x x ≥ 0 Clearly f ( x ) is differentiable at all points.

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