https://r2.erweima.ai/mathbot/image/1725233336077p707o6er.jpg

resuelve el siguiente ejercicio y grafica:

### Solution Overview:

1. **Derivative Calculation**:
   The derivative of the function is:
   \[
   f'(x) = \frac{x(x - 4)}{(1 - x^2)^{\frac{3}{2}}} + \frac{1}{\sqrt{1 - x^2}}
   \]
   This derivative is used to find where the function is increasing or decreasing.

2. **Critical Point**:
   By solving \(f'(x) = 0\), we found that the critical point is \(x = \frac{1}{4}\).

3. **Intervals of Increase and Decrease**:
   - To determine where the function is increasing or decreasing, evaluate the sign of \(f'(x)\) in the intervals determined by the critical point and the domain.
   - \(f'(x)\) changes its sign around \(x = \frac{1}{4}\), which means the function changes from increasing to decreasing at this point.

4. **Graph**:
   The graph of the function \(f(x) = \frac{x-4}{\sqrt{1-x^2}}\) has been plotted. The graph shows a vertical asymptote at \(x = \pm 1\), and the function is defined only within the interval \((-1, 1)\).

### Additional Questions:
1. How does the function behave as \(x\) approaches \(-1\) and \(1\)?
2. What is the significance of the critical point \(x = \frac{1}{4}\) in the context of the graph?
3. How can you confirm the intervals of increase and decrease using the second derivative test?
4. Can this function have any maximum or minimum values within the interval \((-1, 1)\)?
5. What happens to the function as \(x\) approaches the critical points \(x = \pm 1\)?

### Tip:
When dealing with functions involving square roots in the denominator, always check for potential undefined points by examining the expression inside the square root.

Learn how to solve and graph the function f(x) = (x-4) / sqrt(1-x^2). This problem involves calculus concepts such as derivatives and critical points. Includes a detailed step-by-step solution and graphical representation. Suitable for college-level students.

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