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Condition for Coplanarity of Four Points

If f:R→ R b...

Question

If $f : R \to R$ be a differentiable function, such that $f (x + 2 y) = f (x) + f (2 y) + 4 x y \forall x, y ϵ R,$ then

f^{'} (1) = f^{'} (0) + 1

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f^{'} (1) = f^{'} (0) - 1

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f^{'} (0) = f^{'} (1) + 2

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f^{'} (0) = f^{'} (1) - 2

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Solution

The correct option is A $f^{'} (0) = f^{'} (1) - 2$
$f (x + 2 y) = f (x) + f (2 y) + 4 x y \forall x, y ϵ R,$
$\frac{f (x + 2 y) - f (x)}{2 y} = \frac{f (2 y)}{2 y} + 2 x$
${lim}_{2 y \to 0} \frac{f (x + 2 y) - f (x)}{2 y} = {lim}_{2 y \to 0} \frac{f (2 y)}{2 y} + 2 x$
If limit exits for all values of x and y, then ${lim}_{2 y \to 0} \frac{f (2 y)}{2 y} = k$
Thus $f^{'} (x) = k + 2 x$
$f^{'} (1) - f^{'} (0) = 2$

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