consider the function f(X) =x+cosx
which of the following is not true about y =f(x)?

Options

A. It is an increasing function
B. It is a monotonic function
C. It has infinite points of inflection s
D. None of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = x + \cos x \) and determine which of the given statements about this function is not true.

### Step 1: Find the first derivative of the function
To understand the behavior of the function, we first calculate its derivative:

\[ f'(x) = \frac{d}{dx}(x + \cos x) = 1 - \sin x \]

**Hint:** The derivative of a function gives us information about its increasing or decreasing nature.

### Step 2: Analyze the first derivative
Next, we need to analyze the first derivative \( f'(x) = 1 - \sin x \).

- The sine function, \( \sin x \), oscillates between -1 and 1.
- Therefore, \( 1 - \sin x \) will oscillate between \( 1 - 1 = 0 \) and \( 1 - (-1) = 2 \).

This means that \( f'(x) \geq 0 \) for all \( x \). Specifically, \( f'(x) = 0 \) when \( \sin x = 1 \), which occurs at \( x = \frac{\pi}{2} + 2n\pi \) for any integer \( n \).

**Hint:** A derivative that is non-negative indicates that the function is either increasing or constant.

### Step 3: Determine the monotonicity of the function
Since \( f'(x) \) is non-negative, we conclude that the function \( f(x) \) is non-decreasing.

- When \( f'(x) = 0 \), the function is constant at those points, but otherwise, it is increasing.

**Hint:** A function that is non-decreasing is considered monotonic.

### Step 4: Check for points of inflection
The points where \( f'(x) = 0 \) are potential points of inflection or local maxima/minima. However, since \( f'(x) \) does not change sign (it is either 0 or positive), the function does not have any local maxima or minima, but it does have points where it is flat.

**Hint:** Points of inflection occur where the first derivative is zero and changes sign.

### Step 5: Conclusion about the statements
Given the analysis, we can summarize the properties of the function \( f(x) = x + \cos x \):

1. The function is increasing.
2. The function is monotonic.
3. The function has infinite points where it is flat (where \( f'(x) = 0 \)).

Now, we need to identify which of the provided statements is not true. Based on our analysis, if all statements regarding the function's behavior are true, then the answer to the question is that none of the statements are false.

### Final Answer
The answer to the question is option D: None of these.

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