Find the interval in which the function `f(x)=5x^((3)/(2))-3x^((5)/(2)), x gt 0` is strictly decreasing.

Answer

To find the interval in which the function \( f(x) = 5x^{\frac{3}{2}} - 3x^{\frac{5}{2}} \) is strictly decreasing for \( x > 0 \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

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

Using the power rule for differentiation:

\[ f'(x) = 5 \cdot \frac{3}{2} x^{\frac{3}{2} - 1} - 3 \cdot \frac{5}{2} x^{\frac{5}{2} - 1} \]

This simplifies to:

\[ f'(x) = \frac{15}{2} x^{\frac{1}{2}} - \frac{15}{2} x^{\frac{3}{2}} \]

Factoring out \( \frac{15}{2} x^{\frac{1}{2}} \):

\[ f'(x) = \frac{15}{2} x^{\frac{1}{2}} (1 - x) \]

### Step 2: Determine where the derivative is less than zero
To find where the function is strictly decreasing, we need to set the derivative less than zero:

\[ f'(x) < 0 \]

This leads to:

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

Since \( \frac{15}{2} x^{\frac{1}{2}} \) is positive for \( x > 0 \), we focus on the term \( (1 - x) \):

\[ 1 - x < 0 \implies x > 1 \]

### Step 3: Conclusion
Thus, the function \( f(x) \) is strictly decreasing for:

\[ x > 1 \]

### Final Answer
The interval in which the function is strictly decreasing is \( (1, \infty) \).

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