For what values of 'x' are the following functions increasing or decreasing?
`y=5x^(3//2)-3x^(5//2), x gt0`

Answer

To determine the values of \( x \) for which the function \( y = 5x^{3/2} - 3x^{5/2} \) is increasing or decreasing, we need to follow these steps:

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

\[ y = 5x^{3/2} - 3x^{5/2} \]

Using the power rule for differentiation, we find:

\[ \frac{dy}{dx} = \frac{d}{dx}(5x^{3/2}) - \frac{d}{dx}(3x^{5/2}) \]

Calculating each term:

\[ \frac{d}{dx}(5x^{3/2}) = 5 \cdot \frac{3}{2} x^{3/2 - 1} = \frac{15}{2} x^{1/2} \]

\[ \frac{d}{dx}(3x^{5/2}) = 3 \cdot \frac{5}{2} x^{5/2 - 1} = \frac{15}{2} x^{3/2} \]

Thus, the derivative is:

\[ \frac{dy}{dx} = \frac{15}{2} x^{1/2} - \frac{15}{2} x^{3/2} \]

### Step 2: Factor the derivative
Now, we can factor out \( \frac{15}{2} x^{1/2} \):

\[ \frac{dy}{dx} = \frac{15}{2} x^{1/2} (1 - x) \]

### Step 3: Determine where the derivative is zero or undefined
To find the critical points, we set the derivative equal to zero:

\[ \frac{15}{2} x^{1/2} (1 - x) = 0 \]

This gives us two cases:

1. \( x^{1/2} = 0 \) which implies \( x = 0 \) (but we are only considering \( x > 0 \)).
2. \( 1 - x = 0 \) which implies \( x = 1 \).

### Step 4: Analyze the sign of the derivative
Now we need to analyze the sign of \( \frac{dy}{dx} \) in the intervals determined by the critical point \( x = 1 \).

- **Interval 1:** \( (0, 1) \)
- Choose a test point, e.g., \( x = 0.5 \):
\[ \frac{dy}{dx} = \frac{15}{2} (0.5)^{1/2} (1 - 0.5) = \frac{15}{2} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2} > 0 \]
Thus, \( y \) is increasing in this interval.

- **Interval 2:** \( (1, \infty) \)
- Choose a test point, e.g., \( x = 2 \):
\[ \frac{dy}{dx} = \frac{15}{2} (2)^{1/2} (1 - 2) = \frac{15}{2} \cdot \sqrt{2} \cdot (-1) < 0 \]
Thus, \( y \) is decreasing in this interval.

### Conclusion
The function \( y = 5x^{3/2} - 3x^{5/2} \) is:
- **Increasing** on the interval \( (0, 1) \)
- **Decreasing** on the interval \( (1, \infty) \)