(a) Using a graph to find the absolute maximum and minimum values

The function given is:

$f(x) = 3x - 6\cos(x)$

and we are asked to find the absolute maximum and minimum values on the interval $[-2, 0]$. Graphing the function will give us an idea of the extreme points.

Based on the graph, the approximate maximum and minimum values are:

• Maximum: Approximately occurs near $x \approx -0.79$, with $f(x) \approx 4.29$.
• Minimum: Occurs at $x = -2$, with $f(x) \approx -12.39$.

Now, let’s proceed to the exact solution using calculus.

(b) Using calculus to find the exact maximum and minimum values

Step 1: Find the derivative of the function $f(x)$

The first step in finding the extrema is to compute the derivative of $f(x)$:

$f'(x) = \frac{d}{dx} \left( 3x - 6\cos(x) \right)$

$f'(x) = 3 + 6\sin(x)$

Step 2: Set $f'(x) = 0$ to find critical points

Now, set the derivative equal to zero to find the critical points:

$3 + 6\sin(x) = 0$

$\sin(x) = -\frac{1}{2}$

The solutions to $\sin(x) = -\frac{1}{2}$ in the interval $[-2, 0]$ are:

$x = -\frac{\pi}{6}, -\frac{7\pi}{6}$

Since $-\frac{7\pi}{6}$ is outside the given interval, we discard it. Therefore, we only have:

$x = -\frac{\pi}{6}$

Step 3: Evaluate $f(x)$ at critical points and endpoints

Now we evaluate $f(x)$ at the critical point and the endpoints $x = -2$ and $x = 0$.

• At $x = -\frac{\pi}{6}$:

$f\left(-\frac{\pi}{6}\right) = 3\left(-\frac{\pi}{6}\right) - 6\cos\left(-\frac{\pi}{6}\right)$

$f\left(-\frac{\pi}{6}\right) = -\frac{\pi}{2} - 6\left(\frac{\sqrt{3}}{2}\right)$

$f\left(-\frac{\pi}{6}\right) = -\frac{\pi}{2} - 3\sqrt{3} \approx -7.27$

• At $x = -2$:

$f(-2) = 3(-2) - 6\cos(-2)$

$f(-2) = -6 - 6\cos(2)$

Using $\cos(2) \approx -0.4161$:

$f(-2) = -6 - 6(-0.4161) = -6 + 2.4966 \approx -3.5034$

• At $x = 0$:

$f(0) = 3(0) - 6\cos(0) = -6$

Step 4: Compare values to find maximum and minimum

• $f(-\frac{\pi}{6}) \approx -7.27$
• $f(-2) \approx -3.50$
• $f(0) = -6$

Thus, the maximum value is $f(-2) \approx -3.50$ and the minimum value is $f(-\frac{\pi}{6}) \approx -7.27$.

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Conclusion:

• Maximum value: $f(-2) = -3.50$
• Minimum value: $f(-\frac{\pi}{6}) \approx -7.27$

Would you like to explore further details or ask any questions?

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Here are 5 relative questions for further exploration:

1. How would the extrema change if the interval was shifted?
2. What happens to the critical points if the function involves higher-order trigonometric terms?
3. Can you find the second derivative to classify the critical points as maximum or minimum?
4. How does the cosine term affect the overall shape of the function?
5. Could you use numerical methods to verify the extrema more precisely?

Tip: When finding critical points, always check both endpoints and the interior points where the derivative is zero.