Suppose a dif...

Question

Suppose a differentiable function $f (x)$ satisfies the identity $f (x + y) = f (x) + f (y) + x y^{2} + x^{2} y$ for all real $x$ and $y .$ If $lim x \to 0 \frac{f (x)}{x} = 1$ , then $f^{'} (3)$ is equal to

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Solution

Given: $f (x + y) = f (x) + f (y) + x y^{2} + x^{2} y$
At $x = y = 0$
$f (0) = 2 f (0) \Rightarrow f (0) = 0$
Now,
$f^{'} (x) = lim h \to 0 \frac{f (x + h) - f (x)}{h}$
(take $y = h$ )
$f^{'} (x) = lim h \to 0 \frac{f (h)}{h} + lim h \to 0 (x h) + x^{2}$
$f^{'} (x) = 1 + 0 + x^{2}$
$f^{'} (x) = 1 + x^{2}$
$f^{'} (3) = 10$

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