If the function `f(x) = 7/x -3, x in R, x ne 0`, is decreasing function then `x in `……………

Answer

To determine the intervals where the function \( f(x) = \frac{7}{x} - 3 \) is decreasing, we need to find its derivative and analyze the sign of the derivative.

### Step-by-Step Solution:

1. **Find the Derivative**:
The first step is to differentiate the function \( f(x) \).
\[ f'(x) = \frac{d}{dx}\left(\frac{7}{x} - 3\right) \]
The derivative of \( \frac{7}{x} \) can be computed using the power rule:
\[ f'(x) = -\frac{7}{x^2} \]
The derivative of the constant \(-3\) is \(0\).

2. **Set the Derivative Less Than Zero**:
For the function to be decreasing, we need:
\[ f'(x) < 0 \]
Substituting the derivative we found:
\[ -\frac{7}{x^2} < 0 \]

3. **Analyze the Inequality**:
Since \(-\frac{7}{x^2}\) is negative when \(x^2\) is positive, we can multiply both sides of the inequality by \(-1\) (which reverses the inequality):
\[ \frac{7}{x^2} > 0 \]

4. **Determine the Values of \(x\)**:
The expression \(\frac{7}{x^2}\) is positive for all \(x\) except \(x = 0\) (since division by zero is undefined). Thus, we conclude:
\[ x^2 > 0 \quad \text{for } x \in \mathbb{R}, x \neq 0 \]

5. **Final Conclusion**:
Therefore, the function \( f(x) \) is decreasing for:
\[ x \in \mathbb{R}, x \neq 0 \]