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Let's go through the questions one by one.

### 1. **Critical Points** of the Function:
The given function is \( f(x) = \frac{2x + 4}{x^2 + 4} \).

To find the critical points, we first need to compute the derivative of \( f(x) \) using the quotient rule:
\[
f'(x) = \frac{(x^2 + 4)(2) - (2x + 4)(2x)}{(x^2 + 4)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{2(x^2 + 4) - 2x(2x + 4)}{(x^2 + 4)^2}
= \frac{2x^2 + 8 - (4x^2 + 8x)}{(x^2 + 4)^2}
= \frac{2x^2 + 8 - 4x^2 - 8x}{(x^2 + 4)^2}
= \frac{-2x^2 - 8x + 8}{(x^2 + 4)^2}
\]
So, the derivative is:
\[
f'(x) = \frac{-2(x^2 + 4x - 4)}{(x^2 + 4)^2}
\]

To find the critical points, set \( f'(x) = 0 \), which implies solving \( -2(x^2 + 4x - 4) = 0 \):
\[
x^2 + 4x - 4 = 0
\]
Using the quadratic formula to solve for \( x \):
\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-4)}}{2(1)} = \frac{-4 \pm \sqrt{16 + 16}}{2} = \frac{-4 \pm \sqrt{32}}{2} = \frac{-4 \pm 4\sqrt{2}}{2} = -2 \pm 2\sqrt{2}
\]

So, the critical points are:
\[
x = -2 \pm 2\sqrt{2}
\]

### 2. **Local Minimum and Maximum**:
To determine whether these critical points yield a local minimum or maximum, we use the First Derivative Test.

The critical points are \( x = -2 - 2\sqrt{2} \) and \( x = -2 + 2\sqrt{2} \).

From the results:
- At \( x = -2 - 2\sqrt{2} \), this corresponds to a **local minimum**.
- At \( x = -2 + 2\sqrt{2} \), this corresponds to a **local maximum**.

### 3. **Increasing and Decreasing Intervals**:
To find the intervals on which the function is increasing or decreasing, we look at the sign of \( f'(x) \) around the critical points.

- The function is **increasing** on the interval \( (-\infty, -2 - 2\sqrt{2}) \cup (-2 + 2\sqrt{2}, \infty) \).
- The function is **decreasing** on the interval \( (-2 - 2\sqrt{2}, -2 + 2\sqrt{2}) \).

If you'd like further details, feel free to ask!

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Here are 5 related questions that might help you understand the problem more:

1. How do you apply the First Derivative Test for increasing and decreasing intervals?
2. What is the quotient rule for differentiation?
3. How do critical points help identify maxima and minima?
4. How do you interpret intervals in the context of a function's behavior?
5. How do you solve quadratic equations using the quadratic formula?

### Tip:
When solving for critical points, always remember to verify the nature of the points (maxima, minima) using either the second derivative test or the first derivative test.

Consider the function f(x) = (2x + 4) / (x^2 + 4). Find the critical points, determine local maxima and minima, and identify the intervals of increase and decrease.

This calculus problem involves finding the critical points, local maxima and minima, and the intervals of increase and decrease for the function f(x) = (2x + 4) / (x^2 + 4). The solution uses the first derivative test and explores key concepts in differential calculus.

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