Determine for which values of x, the function `y=x^(4)-(4x^(3))/(3)` is increasing and for which it is decreasing.

Answer

To determine the intervals where the function \( y = x^4 - \frac{4x^3}{3} \) is increasing or decreasing, we need to follow these steps:

### Step 1: Find the derivative of the function
We start by differentiating the function with respect to \( x \).

\[ \frac{dy}{dx} = \frac{d}{dx}\left(x^4 - \frac{4}{3}x^3\right) \]

Using the power rule of differentiation:

\[ \frac{dy}{dx} = 4x^3 - 4x^2 \]

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

\[ \frac{dy}{dx} = 4x^2(x - 1) \]

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

\[ 4x^2(x - 1) = 0 \]

This gives us:

\[ 4x^2 = 0 \quad \text{or} \quad x - 1 = 0 \]

From this, we find:

\[ x = 0 \quad \text{and} \quad x = 1 \]

### Step 4: Determine the sign of the derivative
We will use the critical points to test the intervals in which the derivative is positive or negative. The critical points divide the number line into intervals:

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

We will choose test points from each interval to determine the sign of \( \frac{dy}{dx} \).

- **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \)
\[ \frac{dy}{dx} = 4(-1)^2(-1 - 1) = 4(1)(-2) = -8 \quad (\text{Negative}) \]

- **Interval \( (0, 1) \)**: Choose \( x = 0.5 \)
\[ \frac{dy}{dx} = 4(0.5)^2(0.5 - 1) = 4(0.25)(-0.5) = -0.5 \quad (\text{Negative}) \]

- **Interval \( (1, \infty) \)**: Choose \( x = 2 \)
\[ \frac{dy}{dx} = 4(2)^2(2 - 1) = 4(4)(1) = 16 \quad (\text{Positive}) \]

### Step 5: Determine intervals of increase and decrease
From the sign of the derivative:

- The function is **decreasing** on the intervals \( (-\infty, 0) \) and \( (0, 1) \) since \( \frac{dy}{dx} < 0 \).
- The function is **increasing** on the interval \( (1, \infty) \) since \( \frac{dy}{dx} > 0 \).

### Final Answer:
- The function \( y = x^4 - \frac{4x^3}{3} \) is **increasing** for \( x \in (1, \infty) \).
- The function is **decreasing** for \( x \in (-\infty, 0) \) and \( x \in (0, 1) \).