Let `f:R to R` be a function satisfying `f(x+y)=f(x)+f(y)"for all "x,y in R`
`"If "f(x)=x^(3)g(x)"for all "x,yin R`, where g(x) is continuous, then f'(x) is equal to

Options

A. g(0)
B. g'(x)
C. 0
D. none of these

Explanation

To solve the problem step by step, we will use the properties of the function \( f \) and the definition of the derivative.

### Step 1: Understanding the function
We are given that \( f: \mathbb{R} \to \mathbb{R} \) satisfies the functional equation:
\[ f(x+y) = f(x) + f(y) \quad \text{for all } x, y \in \mathbb{R} \]
This implies that \( f \) is a linear function.

### Step 2: Expressing \( f(x) \)
We are also given that:
\[ f(x) = x^3 g(x) \]
where \( g(x) \) is a continuous function.

### Step 3: Finding the derivative \( f'(x) \)
To find \( f'(x) \), we will use the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

### Step 4: Calculating \( f(x+h) \)
Using the functional equation:
\[ f(x+h) = f(x) + f(h) \]
Thus, we can rewrite the derivative as:
\[ f'(x) = \lim_{h \to 0} \frac{f(x) + f(h) - f(x)}{h} = \lim_{h \to 0} \frac{f(h)}{h} \]

### Step 5: Substituting \( f(h) \)
Now substituting \( f(h) = h^3 g(h) \):
\[ f'(x) = \lim_{h \to 0} \frac{h^3 g(h)}{h} = \lim_{h \to 0} h^2 g(h) \]

### Step 6: Evaluating the limit
As \( h \to 0 \), \( h^2 \to 0 \) and \( g(h) \) is continuous, thus \( g(h) \to g(0) \). Therefore:
\[ f'(x) = \lim_{h \to 0} h^2 g(h) = 0 \cdot g(0) = 0 \]

### Conclusion
Thus, we find that:
\[ f'(x) = 0 \]

### Final Answer
The derivative \( f'(x) \) is equal to \( 0 \).

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