Determine for which values of x, the following functions are increasing or decreasing :
`f(x)=x^(4)-2x^(2)`

Answer

To determine the values of \( x \) for which the function \( f(x) = x^4 - 2x^2 \) is increasing or decreasing, we will follow these steps:

### Step 1: Find the derivative of the function
We start by calculating the first derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx}(x^4 - 2x^2) \]

Using the power rule, we differentiate each term:

\[ f'(x) = 4x^3 - 4x \]

### Step 2: Factor the derivative
Next, we factor the derivative to find critical points.

\[ f'(x) = 4x(x^2 - 1) \]

We can further factor \( x^2 - 1 \) as a difference of squares:

\[ f'(x) = 4x(x - 1)(x + 1) \]

### Step 3: Find critical points
To find the critical points, we set the derivative equal to zero:

\[ 4x(x - 1)(x + 1) = 0 \]

This gives us the critical points:

\[ x = 0, \quad x = 1, \quad x = -1 \]

### Step 4: Determine intervals for testing
We will test the sign of \( f'(x) \) in the intervals determined by the critical points. The intervals are:

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

### Step 5: Test the sign of \( f'(x) \) in each interval
We will choose test points from each interval:

- For \( x = -2 \) in \( (-\infty, -1) \):
\[ f'(-2) = 4(-2)((-2) - 1)((-2) + 1) = 4(-2)(-3)(-1) = -24 \quad (\text{Negative}) \]

- For \( x = -0.5 \) in \( (-1, 0) \):
\[ f'(-0.5) = 4(-0.5)((-0.5) - 1)((-0.5) + 1) = 4(-0.5)(-1.5)(0.5) = 1.5 \quad (\text{Positive}) \]

- For \( x = 0.5 \) in \( (0, 1) \):
\[ f'(0.5) = 4(0.5)((0.5) - 1)((0.5) + 1) = 4(0.5)(-0.5)(1.5) = -1.5 \quad (\text{Negative}) \]

- For \( x = 2 \) in \( (1, \infty) \):
\[ f'(2) = 4(2)((2) - 1)((2) + 1) = 4(2)(1)(3) = 24 \quad (\text{Positive}) \]

### Step 6: Summarize the intervals
From our tests, we can summarize the behavior of \( f'(x) \):

- \( f'(x) < 0 \) in \( (-\infty, -1) \) (decreasing)
- \( f'(x) > 0 \) in \( (-1, 0) \) (increasing)
- \( f'(x) < 0 \) in \( (0, 1) \) (decreasing)
- \( f'(x) > 0 \) in \( (1, \infty) \) (increasing)

### Conclusion
The function \( f(x) = x^4 - 2x^2 \) is:
- Decreasing on the intervals \( (-\infty, -1) \) and \( (0, 1) \)
- Increasing on the intervals \( (-1, 0) \) and \( (1, \infty) \)