For what values of 'x' are the following functions increasing or decreasing?
`y=x+(1)/(x), xne0`

Answer

To determine the values of \( x \) for which the function \( y = x + \frac{1}{x} \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
The first step is to differentiate the function with respect to \( x \).

\[ y = x + \frac{1}{x} \]

Differentiating \( y \):

\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]

### Step 2: Set the derivative equal to zero to find critical points
Next, we need to find the critical points by setting the derivative equal to zero.

\[ 1 - \frac{1}{x^2} = 0 \]

Solving for \( x \):

\[ \frac{1}{x^2} = 1 \implies x^2 = 1 \implies x = \pm 1 \]

### Step 3: Determine intervals for testing
The critical points divide the number line into intervals. The critical points are \( x = -1 \) and \( x = 1 \). We will test the intervals:

1. \( (-\infty, -1) \)
2. \( (-1, 0) \)
3. \( (0, 1) \)
4. \( (1, \infty) \)

### Step 4: Test the sign of the derivative in each interval
We will choose test points from each interval to determine the sign of \( \frac{dy}{dx} \).

1. For \( x = -2 \) (in \( (-\infty, -1) \)):
\[ \frac{dy}{dx} = 1 - \frac{1}{(-2)^2} = 1 - \frac{1}{4} = \frac{3}{4} > 0 \quad \text{(increasing)} \]

2. For \( x = -0.5 \) (in \( (-1, 0) \)):
\[ \frac{dy}{dx} = 1 - \frac{1}{(-0.5)^2} = 1 - 4 = -3 < 0 \quad \text{(decreasing)} \]

3. For \( x = 0.5 \) (in \( (0, 1) \)):
\[ \frac{dy}{dx} = 1 - \frac{1}{(0.5)^2} = 1 - 4 = -3 < 0 \quad \text{(decreasing)} \]

4. For \( x = 2 \) (in \( (1, \infty) \)):
\[ \frac{dy}{dx} = 1 - \frac{1}{(2)^2} = 1 - \frac{1}{4} = \frac{3}{4} > 0 \quad \text{(increasing)} \]

### Step 5: Summarize the intervals
From our tests, we can summarize the behavior of the function:

- The function is **increasing** on the intervals \( (-\infty, -1) \) and \( (1, \infty) \).
- The function is **decreasing** on the intervals \( (-1, 0) \) and \( (0, 1) \).

### Final Result
- **Increasing**: \( (-\infty, -1) \) and \( (1, \infty) \)
- **Decreasing**: \( (-1, 0) \) and \( (0, 1) \)