To determine the absolute extrema of the function $f(x) = \frac{x - 2}{x^2 + 3x + 6}$, we need to:

1. Find the critical points by taking the derivative of $f(x)$ and setting it to zero.
2. Evaluate $f(x)$ at critical points and endpoints (if applicable) to find extrema.
3. Analyze the behavior of the function to see if extrema exist.

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Step 1: Identify Critical Points

The derivative of $f(x)$ can be calculated using the quotient rule: $f'(x) = \frac{(x^2 + 3x + 6)(1) - (x - 2)(2x + 3)}{(x^2 + 3x + 6)^2}.$

Simplify the numerator: $\text{Numerator} = (x^2 + 3x + 6) - (2x^2 + 7x - 6) = -x^2 - 4x + 12.$

Thus: $f'(x) = \frac{-x^2 - 4x + 12}{(x^2 + 3x + 6)^2}.$

Set the numerator equal to 0 to find critical points: $-x^2 - 4x + 12 = 0 \quad \implies \quad x^2 + 4x - 12 = 0.$

Solve the quadratic equation: $x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)} = \frac{-4 \pm \sqrt{16 + 48}}{2} = \frac{-4 \pm \sqrt{64}}{2} = \frac{-4 \pm 8}{2}.$

$x = 2 \quad \text{or} \quad x = -6.$

Step 2: Evaluate $f(x)$ at Critical Points

Substitute $x = 2$ and $x = -6$ into $f(x)$: $f(2) = \frac{2 - 2}{2^2 + 3(2) + 6} = \frac{0}{16} = 0.$

$f(-6) = \frac{-6 - 2}{(-6)^2 + 3(-6) + 6} = \frac{-8}{36 - 18 + 6} = \frac{-8}{24} = -\frac{1}{3}.$

Step 3: Analyze the End Behavior

The denominator $x^2 + 3x + 6$ is always positive, so $f(x)$ is continuous for all $x$.

As $x \to \pm\infty$, $f(x) \to 0$ because the degree of the numerator is less than the degree of the denominator.

Step 4: Determine Absolute Extrema

Now compare the function values at the critical points:

• $f(-6) = -\frac{1}{3}$
• $f(2) = 0$

The absolute minimum is $-\frac{1}{3}$ at $x = -6$, and the function has no absolute maximum.

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Correct Answer:

C. No absolute extrema.

Would you like a detailed explanation of any step?

5 Related Questions:

1. How is the quotient rule applied to derivatives?
2. Why is analyzing end behavior important for finding extrema?
3. How do you solve a quadratic equation using the discriminant?
4. What are the conditions for a function to have absolute extrema?
5. Why does the degree of the numerator and denominator determine the behavior at infinity?

Tip:

When analyzing a rational function, always simplify and check continuity to avoid missing critical points.