Let f be a function such that f(x + y) = f(x) | vedclass

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(A) Given $f(x + y) = f(x) + f(y)$,this is Cauchy's functional equation,which implies $f(x) = cx$ for some constant $c$.Using the definition of the de

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$9$

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(A) Given $f(x + y) = f(x) + f(y)$,this is Cauchy's functional equation,which implies $f(x) = cx$ for some constant $c$.Using the definition of the derivative: $f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h} = \lim_{h 	o 0} \frac{f(h)}{h}$.Given $f(x) = (2x^2 + 3x)g(x)$,we have $f(h) = (2h^2 + 3h)g(h)$.Substituting this into the derivative formula: $f'(x) = \lim_{h 	o 0} \frac{(2h^2 + 3h)g(h)}{h} = \lim_{h 	o 0} (2h + 3)g(h)$.Since $g(x)$ is continuous at $x = 0$,$\lim_{h 	o 0} g(h) = g(0) = 3$.Thus,$f'(x) = (2(0) + 3) 	imes 3 = 3 	imes 3 = 9$.

Let $f$ be a function such that $f(x + y) = f(x) + f(y)$ for all $x$ and $y$,and $f(x) = (2x^2 + 3x)g(x)$ for all $x$,where $g(x)$ is continuous and $g(0) = 3$. Then $f'(x)$ is equal to:

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