Let f be a function satisfying f(x+y)=f(x)+f(y) and f(x)=x^3 ϕ(x) for all x and y, where ϕ(x) is

step 1 In this problem, we can write F -dash x as equals to limit H tends to 0, F of x plus H minus F o

Let f be a function satisfying f(x+y)=f(x)+f(y) and f(x)=x^3...

Let $f$ be a function satisfying $f(x+y)=f(x)+f(y)$ and $f(x)=x^{3} \phi(x)$ for all $x$ and $y$, where $\phi(x)$ is a continuous function then $f^{\prime}(x)$ is equal to
(A) $g(0)$
(B) $g^{\prime}(x)$
(C) 0
(D) None of these

Key Concepts

Cauchy's Functional Equation

This is an equation where a function f satisfies f(x+y)=f(x)+f(y). Under the additional assumption of continuity (or other regularity conditions), the solutions are forced to be linear functions. This concept is fundamental in understanding how additive properties can determine the structure of a function.

Continuity Leading to Linearity

When continuity is imposed on the solution of Cauchy’s functional equation, any 'exotic' solutions are ruled out, and the function must be linear. In this problem, the additional representation f(x)=x³?(x) and the continuity of ? ensure that the only possibility is for the linear coefficient to be zero, showing how continuity guarantees standard behavior.

Consistency of Functional Representations

The function is expressed in two different forms: one as a solution to an additive equation and the other as a product involving a continuous function. Analyzing how these forms coexist under the given conditions highlights the importance of ensuring that different representations of a function remain consistent, often leading to further constraints on the function's parameters.

Derivative of a Constant (Zero) Function

Once it is determined that the function must be identically zero, it follows that its derivative is also zero. Understanding this basic fact about differentiability is crucial when analyzing functions that, due to imposed conditions, reduce to trivial or constant forms.

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