To find the absolute extrema of the function $y = 7 \cos(x)$ on the closed interval $[0, 2\pi]$, we'll go through the following steps:

Step 1: Identify the function

The function is given as: $y = 7 \cos(x)$

Step 2: Find the derivative of $y$

To locate the critical points, we need to take the derivative of $y$ with respect to $x$: $y' = -7 \sin(x)$

Step 3: Set the derivative to zero to find critical points

We set $y' = 0$ to find the critical points within the interval $[0, 2\pi]$: $-7 \sin(x) = 0$ $\sin(x) = 0$ The values of $x$ where $\sin(x) = 0$ in the interval $[0, 2\pi]$ are: $x = 0, \pi, \text{ and } 2\pi$

Step 4: Evaluate $y$ at the critical points and endpoints

Now, we evaluate $y = 7 \cos(x)$ at $x = 0$, $x = \pi$, and $x = 2\pi$.

1. At $x = 0$: $y = 7 \cos(0) = 7 \times 1 = 7$

2. At $x = \pi$: $y = 7 \cos(\pi) = 7 \times (-1) = -7$

3. At $x = 2\pi$: $y = 7 \cos(2\pi) = 7 \times 1 = 7$

Step 5: Determine the absolute extrema

• The maximum value of $y$ is 7, which occurs at both $x = 0$ and $x = 2\pi$.
• The minimum value of $y$ is -7, which occurs at $x = \pi$.

Final Answer

• Minimum: $(x, y) = (\pi, -7)$
• Maximum (smaller $x$-value): $(x, y) = (0, 7)$
• Maximum (larger $x$-value): $(x, y) = (2\pi, 7)$

Would you like a more detailed explanation, or do you have any questions?

Here are some related questions to expand your understanding:

1. How do we determine critical points for functions involving trigonometric functions?
2. What are the key differences between absolute and relative extrema?
3. Why does $\cos(x)$ have a range of $[-1, 1]$, and how does it affect finding extrema?
4. How would the extrema change if the function were $y = 7 \sin(x)$ instead of $y = 7 \cos(x)$?
5. What is the significance of endpoints in finding absolute extrema on a closed interval?

Tip: When working with trigonometric functions on specific intervals, sketching the function's graph can provide visual insight into where maxima and minima occur.