f(x) is a differentiable function such that f(1/x)=0 for every x belongs to N , then which of the following is/are correct;

A) f(0)=0 C) f(1)=0

B) f'(0)=0 D) f'(1)=0

To tackle the problem regarding the function \( f(x) \) defined by the condition \( f(1/x) = 0 \) for every natural number \( x \), we need to analyze what this implies about the function at specific points, particularly \( f(0) \), \( f(1) \), \( f'(0) \), and \( f'(1) \).

Understanding the Function's Properties

Given that \( f(1/x) = 0 \) for all \( x \in \mathbb{N} \), we can substitute various natural numbers into this equation:

• For \( x = 1 \): \( f(1/1) = f(1) = 0 \)
• For \( x = 2 \): \( f(1/2) = 0 \)
• For \( x = 3 \): \( f(1/3) = 0 \)
• And so on for any natural number \( x \).

This tells us that \( f(1) = 0 \) and \( f(1/n) = 0 \) for all natural numbers \( n \). As \( n \) increases, \( 1/n \) approaches \( 0 \). Since \( f \) is differentiable, it is also continuous, which leads us to consider the behavior of \( f(x) \) as \( x \) approaches \( 0 \).

Evaluating \( f(0) \)

Since \( f(1/n) = 0 \) for all \( n \), and \( 1/n \) approaches \( 0 \) as \( n \) becomes very large, the continuity of \( f \) implies:

Thus, we conclude that:

Since \( f(1/n) \to f(0) \) as \( n \to \infty \), and since \( f(1/n) = 0 \), it follows that \( f(0) = 0 \).

Considering the Derivative at Specific Points

Next, we need to analyze the derivatives \( f'(0) \) and \( f'(1) \).

Finding \( f'(0) \)

The derivative \( f'(0) \) can be evaluated using the limit definition of the derivative:

Using the limit:

f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h)}{h}

Since \( f(h) = 0 \) for all \( h = 1/n \) (where \( n \) is a natural number), and as \( h \) approaches \( 0 \), we can see that \( f(h) \) remains \( 0 \) for these values. Hence, the limit evaluates to:

Therefore:

f'(0) = 0.

Examining \( f'(1) \)

Now, let's look at \( f'(1) \). Since we know \( f(1) = 0 \), we can apply the derivative limit definition again:

f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0} \frac{f(1 + h)}{h}

However, we do not have information about \( f(1 + h) \) for small \( h \). The condition \( f(1/x) = 0 \) does not provide direct insight into values of \( f \) around \( 1 \). Therefore, we cannot definitively conclude that \( f'(1) = 0 \) without additional information about the function's behavior near \( 1 \).

Summarizing the Findings

Based on our analysis, we can conclude the following:

• A) f(0) = 0: True
• B) f'(0) = 0: True
• C) f(1) = 0: True
• D) f'(1) = 0: Cannot be determined from the given information.

In summary, the correct statements are A, B, and C. D cannot be confirmed without further details about the function's behavior near \( 1 \).