Let's break down the solution and explain it step by step.

(a) Domain of $f(x)$

The function $f(x) = \frac{x}{x^2 + 10x + 21}$ is undefined where the denominator equals zero: $x^2 + 10x + 21 = 0$ Factoring: $(x + 7)(x + 3) = 0$ Thus, $x = -7$ and $x = -3$ are the points where the function is undefined. Therefore, the domain of $f(x)$ in interval notation is: $(-\infty, -7) \cup (-7, -3) \cup (-3, \infty)$

(b) Critical Numbers

Critical numbers occur where the derivative of $f(x)$, $f'(x)$, is zero or undefined. Calculating $f'(x)$ using the quotient rule: $f'(x) = \frac{(1)(x^2 + 10x + 21) - x(2x + 10)}{(x^2 + 10x + 21)^2}$ Simplify the numerator: $(x^2 + 10x + 21) - (2x^2 + 10x) = -x^2 + 21$ So, $f'(x) = \frac{-x^2 + 21}{(x^2 + 10x + 21)^2}$ Set the numerator equal to zero to find critical points: $-x^2 + 21 = 0 \quad \Rightarrow \quad x^2 = 21 \quad \Rightarrow \quad x = \pm \sqrt{21}$ Thus, the critical numbers are: $x = -\sqrt{21}, \quad x = \sqrt{21}$

(c) Increasing and Decreasing Intervals

Analyze the sign of $f'(x)$ around the critical points $x = \pm \sqrt{21}$ and where $f(x)$ is undefined ($x = -7, -3$).

1. Test intervals:

   • $(-\infty, -\sqrt{21})$
   • $(-\sqrt{21}, \sqrt{21})$
   • $(\sqrt{21}, \infty)$

2. Sign of $f'(x)$:

   • $f'(x) > 0$ on $(-\sqrt{21}, \sqrt{21})$ (increasing).
   • $f'(x) < 0$ on $(-\infty, -\sqrt{21}) \cup (\sqrt{21}, \infty)$ (decreasing).

So, $f(x)$ is:

• Increasing on $(-\sqrt{21}, \sqrt{21})$.
• Decreasing on $(-\infty, -\sqrt{21}) \cup (\sqrt{21}, \infty)$.

(d) Relative Extrema (First Derivative Test)

1. At $x = -\sqrt{21}$:

   • $f'(x)$ changes from negative to positive.
   • Thus, $x = -\sqrt{21}$ is a relative minimum.

2. At $x = \sqrt{21}$:

   • $f'(x)$ changes from positive to negative.
   • Thus, $x = \sqrt{21}$ is a relative maximum.

Summary:

• Relative maxima: $x = \sqrt{21}$
• Relative minima: $x = -\sqrt{21}$

Recap:

1. Domain: $(-\infty, -7) \cup (-7, -3) \cup (-3, \infty)$
2. Critical numbers: $x = \pm \sqrt{21}$
3. Increasing: $(-\sqrt{21}, \sqrt{21})$
4. Decreasing: $(-\infty, -\sqrt{21}) \cup (\sqrt{21}, \infty)$
5. Relative maxima: $x = \sqrt{21}$
6. Relative minima: $x = -\sqrt{21}$

Do you want a detailed explanation of any specific part? Here are some questions for further exploration:

1. How is the derivative computed using the quotient rule?
2. Why do critical points only occur where $f'(x) = 0$ or undefined?
3. How do we test intervals to determine increasing or decreasing behavior?
4. What happens to $f(x)$ near undefined points ($x = -7, -3$)?
5. Can we verify these results graphically?

Tip: Always factorize carefully and check for undefined points in rational functions.