Let f: R →R be a function satisfying the condition f((x+y)/(k))=(f(x)+f(y))/(k), where k ≠0,2. T

step 1 We are given that F of x plus y by k is equal to fx plus f y by k. Now what we can do is that we

Let f: R →R be a function satisfying the condition...

Let $f: R \rightarrow R$ be a function satisfying the condition $f\left(\frac{x+y}{k}\right)=\frac{f(x)+f(y)}{k}$, where $k \neq 0,2$. The function $f(x)$ is differentiable on $R$ and $f^{\prime}(0)=m$.
$f^{\prime}(x)$ is equal to
(A) $m$
(B) $2 m$
(C) $m+1$
(D) 0

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Key Concepts

Constant Derivative

The concept of a constant derivative is associated with linear functions, which have a uniform rate of change. When a function's derivative is constant, it indicates that the function is linear, and the value of the derivative is the slope of the corresponding line. This also implies that knowing the derivative at one point determines it everywhere.

Functional Equations

Functional equations are equations where the unknowns are functions, and the equations relate the values of these functions at various points. They often lead to determining a specific functional form by exploiting symmetry or invariance properties inherent in the equation.

Linear Functions

Linear functions are functions that can be written in the form f(x) = ax + b. The properties of linearity, such as additivity and homogeneity, make them natural solutions to many functional equations. When additional conditions, like differentiability or specific derivative values, are imposed, the resulting function is often forced into a linear form.

Differentiability

Differentiability of a function means that the function has a derivative at every point in its domain, which is a measure of the instantaneous rate of change. Differentiability is a key concept in calculus, allowing the use of derivative-based methods for function analysis and problem solving.

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