Let `f(x)` be a twice differentiable function for all real values of `x` and satisfies `f(1)=1,f(2)=4,f(3)=9.` Then which of the following is definitely true? (a) `f^('')=2AAx in (1,3)` (b) `f^('')=f(x)=5forsom ex in (2,3)` (c) `f^('')=3AAx in (2,3)` (d) `f^('')=2forsom ex in (1,3)`

Options

A. `f''(x)=2AAx in(1,3)`
B. `f''(x)=f(x)5"for some"x in(2,3)`
C. `f''(x)=3AAx in(2,3)`
D. `f''(x)=2"for some"x in(1,3)`

Answer

To solve the problem, we need to analyze the given conditions and apply Rolle's theorem and the Mean Value Theorem (MVT) appropriately. Let's go through the solution step by step.

### Step 1: Define the Function
Let \( g(x) = f(x) - x^2 \).

### Step 2: Evaluate \( g(x) \) at Given Points
We know:
- \( f(1) = 1 \) implies \( g(1) = f(1) - 1^2 = 1 - 1 = 0 \)
- \( f(2) = 4 \) implies \( g(2) = f(2) - 2^2 = 4 - 4 = 0 \)
- \( f(3) = 9 \) implies \( g(3) = f(3) - 3^2 = 9 - 9 = 0 \)

Thus, we have:
- \( g(1) = 0 \)
- \( g(2) = 0 \)
- \( g(3) = 0 \)

### Step 3: Apply Rolle's Theorem
Since \( g(1) = g(2) = 0 \), by Rolle's theorem, there exists at least one \( c_1 \in (1, 2) \) such that:
\[ g'(c_1) = 0 \]

### Step 4: Differentiate \( g(x) \)
Now, we differentiate \( g(x) \):
\[ g'(x) = f'(x) - 2 \]
Setting this equal to zero gives:
\[ f'(c_1) - 2 = 0 \implies f'(c_1) = 2 \]

### Step 5: Apply Rolle's Theorem Again
Next, since \( g(2) = g(3) = 0 \), we apply Rolle's theorem again in the interval \( (2, 3) \). Thus, there exists at least one \( c_2 \in (2, 3) \) such that:
\[ g'(c_2) = 0 \]
This gives:
\[ f'(c_2) - 2 = 0 \implies f'(c_2) = 2 \]

### Step 6: Differentiate Again
Now we define \( h(x) = g'(x) = f'(x) - 2 \) and differentiate again:
\[ h'(x) = f''(x) \]

### Step 7: Apply Rolle's Theorem Once More
Since \( h(c_1) = 0 \) and \( h(c_2) = 0 \), by Rolle's theorem, there exists a \( c_3 \in (c_1, c_2) \) such that:
\[ h'(c_3) = 0 \implies f''(c_3) = 0 \]

### Conclusion
From the above steps, we conclude that there exists some \( c_3 \in (1, 3) \) such that \( f''(c_3) = 0 \). This means that \( f''(x) = 2 \) for some \( x \) in the interval \( (1, 3) \).

### Final Answer
Thus, the correct option is:
**(d) \( f''(x) = 2 \) for some \( x \in (1, 3) \)**.