Consider, f (x) is a function such that `f(1)=1, f(2)=4 and f(3)=9`
Statement 1 : `f''(x)=2" for some "x in (1,3)`
Statement 2 : `g(x)=x^(2) rArr g''(x)=2, AA x in R`

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze the function \( f(x) \)

We are given that:
- \( f(1) = 1 \)
- \( f(2) = 4 \)
- \( f(3) = 9 \)

From these values, we can observe that:
- \( f(1) = 1^2 \)
- \( f(2) = 2^2 \)
- \( f(3) = 3^2 \)

This suggests that \( f(x) = x^2 \) could be a candidate function.

### Step 2: Determine the first and second derivatives of \( f(x) \)

Now, let's find the first and second derivatives of \( f(x) = x^2 \):
- The first derivative \( f'(x) = 2x \)
- The second derivative \( f''(x) = 2 \)

### Step 3: Verify Statement 1

Statement 1 claims that \( f''(x) = 2 \) for some \( x \) in the interval \( (1, 3) \).

Since we found that \( f''(x) = 2 \) for all \( x \) in \( \mathbb{R} \), it is certainly true for \( x \) in the interval \( (1, 3) \).

Thus, Statement 1 is **true**.

### Step 4: Analyze Statement 2

Statement 2 states that \( g(x) = x^2 \) implies \( g''(x) = 2 \) for all \( x \in \mathbb{R} \).

We already calculated:
- \( g'(x) = 2x \)
- \( g''(x) = 2 \)

This is also true for all \( x \in \mathbb{R} \).

Thus, Statement 2 is also **true**.

### Step 5: Conclusion

Both statements are true:
- Statement 1 is true because \( f''(x) = 2 \) for some \( x \) in \( (1, 3) \).
- Statement 2 is true because \( g''(x) = 2 \) for all \( x \in \mathbb{R} \).

However, the problem states that Statement 2 does not explain Statement 1, which is also correct.

### Final Answer:
Both statements are true, but Statement 2 does not explain Statement 1.

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