For all x in [1, 2] Let `f"(x)` of a non-constant function `f(x)` exist and satisfy `|fprimeprime(x)|<=2.` If `f(1)=f(2)`, then (A) There exist some `a in (1,2)` such that f'(a)=0 (B) f(x) is strictly increasing in (1,2) (C) There exists atleast one `c in (1,2)` such that `f'(c)>0` (D) `|f'(x)| lt 2 AA x in [ 1,2]`

Options

A. There exist some `a in (1,2)` such that f(a)=0
B. f(x) is strictly increasing in (1,2)
C. There exist atleast one `c in (1,2)` such that `f(c) gt 0`
D. `|f'(x)|lt 2 AA x in [1,2]`

Answer

To solve the problem, we will analyze the given conditions and apply Rolle's theorem. Let's go through the solution step by step.

### Step 1: Understand the given conditions
We have a function \( f(x) \) defined on the interval \([1, 2]\) such that:
1. \( f(1) = f(2) \)
2. The second derivative \( f''(x) \) exists and satisfies \( |f''(x)| \leq 2 \).

### Step 2: Apply Rolle's Theorem
Since \( f(1) = f(2) \) and \( f(x) \) is continuous and differentiable on \([1, 2]\), we can apply Rolle's theorem. According to Rolle's theorem, there exists at least one point \( a \) in the interval \( (1, 2) \) such that:
\[ f'(a) = 0 \]
This confirms that option (A) is correct.

### Step 3: Analyze option (B)
We need to determine if \( f(x) \) is strictly increasing on the interval \( (1, 2) \). For \( f(x) \) to be strictly increasing, \( f'(x) \) must be greater than 0 for all \( x \) in \( (1, 2) \). However, since we found a point \( a \) where \( f'(a) = 0 \), it indicates that \( f(x) \) cannot be strictly increasing throughout the entire interval. Therefore, option (B) is incorrect.

### Step 4: Analyze option (C)
We need to check if there exists at least one point \( c \) in \( (1, 2) \) such that \( f'(c) > 0 \). Since \( f'(a) = 0 \) at some point \( a \), and given that \( f''(x) \) is bounded (i.e., \( |f''(x)| \leq 2 \)), the function can change from increasing to decreasing or vice versa. Thus, it is possible for \( f'(x) \) to be greater than 0 at some other point \( c \) in \( (1, 2) \). Therefore, option (C) is correct.

### Step 5: Analyze option (D)
We need to check if \( |f'(x)| < 2 \) for all \( x \) in \([1, 2]\). Since \( |f''(x)| \leq 2 \), this implies that the rate of change of \( f'(x) \) is limited. By integrating \( f''(x) \) over the interval, we can conclude that the maximum change in \( f'(x) \) from one endpoint to the other is bounded. Thus, it is reasonable to conclude that \( |f'(x)| < 2 \) for all \( x \) in \([1, 2]\). Therefore, option (D) is also correct.

### Summary of Results
- Option (A) is correct.
- Option (B) is incorrect.
- Option (C) is correct.
- Option (D) is correct.

### Final Answer
The correct options are (A), (C), and (D).

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