Range of the function `f(x)= x cos( 1/x), xgt1`

Options

A. `[cos1+sin1 , infty)`
B. `[1, infty)`
C. `[cos1,1)`
D. `[cos1, infty)`

Answer

To find the range of the function \( f(x) = x \cos\left(\frac{1}{x}\right) \) for \( x \geq 1 \), we will follow these steps:

### Step 1: Find the derivative of the function
To determine if the function is increasing or decreasing, we first need to calculate the derivative \( f'(x) \).

Using the product rule:
\[ f'(x) = \frac{d}{dx}[x] \cdot \cos\left(\frac{1}{x}\right) + x \cdot \frac{d}{dx}\left[\cos\left(\frac{1}{x}\right)\right] \]

The derivative of \( \cos\left(\frac{1}{x}\right) \) is:
\[ \frac{d}{dx}\left[\cos\left(\frac{1}{x}\right)\right] = -\sin\left(\frac{1}{x}\right) \cdot \frac{d}{dx}\left[\frac{1}{x}\right] = -\sin\left(\frac{1}{x}\right) \cdot \left(-\frac{1}{x^2}\right) = \frac{\sin\left(\frac{1}{x}\right)}{x^2} \]

Thus, we have:
\[ f'(x) = \cos\left(\frac{1}{x}\right) + x \cdot \frac{\sin\left(\frac{1}{x}\right)}{x^2} \]
\[ f'(x) = \cos\left(\frac{1}{x}\right) + \frac{\sin\left(\frac{1}{x}\right)}{x} \]

### Step 2: Analyze the sign of the derivative
For \( x \geq 1 \), we know \( \frac{1}{x} \) will be in the interval \( (0, 1] \).

Since \( \cos\left(\frac{1}{x}\right) \) is positive and \( \frac{\sin\left(\frac{1}{x}\right)}{x} \) is also positive for \( x \geq 1 \), we can conclude that:
\[ f'(x) > 0 \quad \text{for } x \geq 1 \]

This means that \( f(x) \) is an increasing function for \( x \geq 1 \).

### Step 3: Find the minimum value of the function
To find the minimum value of \( f(x) \) when \( x = 1 \):
\[ f(1) = 1 \cdot \cos\left(1\right) = \cos(1) \]

### Step 4: Determine the range of the function
Since \( f(x) \) is increasing for \( x \geq 1 \) and the minimum value at \( x = 1 \) is \( f(1) = \cos(1) \), as \( x \) approaches infinity, \( f(x) \) will also approach infinity.

Thus, the range of the function \( f(x) = x \cos\left(\frac{1}{x}\right) \) for \( x \geq 1 \) is:
\[ [\cos(1), \infty) \]

### Final Answer:
The range of the function \( f(x) = x \cos\left(\frac{1}{x}\right) \) for \( x \geq 1 \) is \( [\cos(1), \infty) \).

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