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To find the critical points of the function \( f(x) = \frac{-x^2 - 1}{x^2 - 4} \), we need to determine where the derivative \( f'(x) \) is equal to zero or undefined.

### Step 1: Find the derivative \( f'(x) \)
We'll use the quotient rule, which states:
\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]
where \( g(x) = -x^2 - 1 \) and \( h(x) = x^2 - 4 \).

1. \( g(x) = -x^2 - 1 \), so \( g'(x) = -2x \).
2. \( h(x) = x^2 - 4 \), so \( h'(x) = 2x \).

Now, applying the quotient rule:
\[
f'(x) = \frac{(-2x)(x^2 - 4) - (-x^2 - 1)(2x)}{(x^2 - 4)^2}
\]

### Step 2: Simplify the expression
Expand and simplify the numerator:
\[
f'(x) = \frac{-2x(x^2 - 4) + 2x(x^2 + 1)}{(x^2 - 4)^2}
\]
\[
f'(x) = \frac{-2x^3 + 8x + 2x^3 + 2x}{(x^2 - 4)^2}
\]
\[
f'(x) = \frac{10x}{(x^2 - 4)^2}
\]

### Step 3: Set the derivative equal to zero to find critical points
For \( f'(x) = 0 \):
\[
\frac{10x}{(x^2 - 4)^2} = 0
\]
This equation equals zero when the numerator is zero:
\[
10x = 0 \quad \Rightarrow \quad x = 0
\]

### Step 4: Verify whether the denominator affects critical points
The function is undefined where the denominator \( (x^2 - 4)^2 \) is zero, which occurs at:
\[
x^2 - 4 = 0 \quad \Rightarrow \quad x = \pm 2
\]
These points are not in the domain of the function, so they are not considered critical points.

### Conclusion:
The function \( f(x) \) has a critical point at \( x = 0 \).

Would you like more details or have any questions?

### Related Questions:
1. What are the second derivative test conditions for classifying the critical point at \( x = 0 \)?
2. How would the graph of \( f(x) \) behave near the critical points?
3. What is the behavior of \( f(x) \) as \( x \) approaches \( \pm 2 \)?
4. How do asymptotes influence the analysis of critical points in rational functions?
5. What other methods can be used to find critical points besides differentiation?

### Tip:
Always consider the domain of the function when finding critical points, as some potential points might lie outside the domain and thus aren't valid.

A função f(x) = (-x^2 - 1) / (x^2 - 4) tem um ponto crítico para x igual a:

Explore the critical points of the rational function f(x) = (-x^2 - 1) / (x^2 - 4). Learn how to find where the derivative equals zero and identify undefined points, using the quotient rule and other calculus concepts. Suitable for students in Grades 11-12.

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