I et f be a function such that f(x+y)=f(x)+f(y) for all x and y and f(x)=(2 x^2+3 x) g(x) for al

step 1 F -dash -X equal limit H tends to 0 F x plus H minus F x upon H. Equal limit H tends to 0. And f

I et f be a function such that f(x+y)=f(x)+f(y) for all x...

I et $f$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x$ and $y$ and $f(x)=\left(2 x^{2}+3 x\right) g(x)$ for all $x$ where $g(x)$ is continuous and $g(0)=3 .$ Then $f^{\prime}(x)$ is equal to
(A) 9
(B) 3
(C) 6
(D) nonc

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Key Concepts

Continuity and Linearity

Continuity plays a crucial role in guaranteeing that an additive function is linear. Without the continuity assumption, there exist pathological solutions to the functional equation, but continuity ensures that the function takes the simple form of a linear function, which is essential when analyzing its derivative.

Cauchy’s Functional Equation

This concept involves functions that satisfy the equation f(x + y) = f(x) + f(y) for all x and y. When such functions are also continuous, one can conclude that they are linear, meaning that f(x) = ax for some constant a. This result is a cornerstone in the study of functional equations and real analysis.

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This method is used to compute the instantaneous rate of change and is fundamental in connecting the function’s local behavior with its global properties.

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