To find the value of \( x \) for which the function \( y = x^4 - \frac{4x^3}{3} \) is increasing, we need to follow these steps:
### Step 1: Differentiate the function
To determine where the function is increasing, we first need to find the derivative of \( y \) 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, we differentiate each term:
\[
\frac{dy}{dx} = 4x^3 - 4x^2
\]
### Step 2: Factor the derivative
Next, we can factor the derivative to simplify our analysis:
\[
\frac{dy}{dx} = 4x^2(x - 1)
\]
### Step 3: Set the derivative greater than zero
To find where the function is increasing, we set the derivative greater than zero:
\[
4x^2(x - 1) > 0
\]
### Step 4: Identify critical points
Now, we need to find the critical points by setting the derivative equal to zero:
\[
4x^2(x - 1) = 0
\]
This gives us:
1. \( 4x^2 = 0 \) → \( x = 0 \)
2. \( x - 1 = 0 \) → \( x = 1 \)
So, the critical points are \( x = 0 \) and \( x = 1 \).
### Step 5: Test intervals around critical points
We will test the intervals determined by the critical points \( (-\infty, 0) \), \( (0, 1) \), and \( (1, \infty) \).
- **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \):
\[
4(-1)^2(-1 - 1) = 4 \cdot 1 \cdot (-2) = -8 \quad (\text{Negative})
\]
- **Interval \( (0, 1) \)**: Choose \( x = 0.5 \):
\[
4(0.5)^2(0.5 - 1) = 4 \cdot 0.25 \cdot (-0.5) = -0.5 \quad (\text{Negative})
\]
- **Interval \( (1, \infty) \)**: Choose \( x = 2 \):
\[
4(2)^2(2 - 1) = 4 \cdot 4 \cdot 1 = 16 \quad (\text{Positive})
\]
### Step 6: Conclusion
From our tests, we see that the function is increasing when \( x \) is in the interval \( (1, \infty) \).
Thus, the value of \( x \) for which the function is increasing is:
\[
\boxed{(1, \infty)}
\]
