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Using differentiability and continuity of a function \( f \) which satisfies certain functional equation, we can determine in some cases the function explicitly. E.g. If \( f \) satisfies \( f(x+y) \) \( =f(x) f(y) \) for all \( x, y \in \mathbf{R} \) and \( f(x) \neq 0 \) for any \( x \in \mathbf{R} \) and \( f^{\prime}(0)=1 \) then \( f(x)=e^{x} \).
If \( f^{\prime}(x)=f(x) \) for all \( x \) and \( f^{\prime}(0)=4 \) then \( f(x) \) is equal to
(a) \( 2 e^{2 x} \)
(b) \( e^{4 x} \)
(c) \( x^{4}+4 x^{2}+4 x \)
(d) \( 4 e^{x} \)
\( \mathrm{W} \)
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