To determine the absolute extreme values of the function $f(x) = 5 \cos^2(x)$ on the interval $[0, \pi]$, we need to follow these steps:

Step 1: Find the critical points

The critical points are values of $x$ where $f'(x) = 0$ or where $f'(x)$ is undefined.

1. Rewrite $f(x)$: $f(x) = 5 \cos^2(x)$

2. Differentiate $f(x)$: Using the chain rule, we differentiate $f(x)$ with respect to $x$: $f'(x) = 5 \cdot 2 \cos(x) \cdot (-\sin(x)) = -10 \cos(x) \sin(x)$

   Using the double-angle identity, $\sin(2x) = 2 \sin(x) \cos(x)$, we can simplify this as: $f'(x) = -5 \sin(2x)$

3. Set $f'(x) = 0$ and solve for $x$: $-5 \sin(2x) = 0$ Dividing by $-5$, we get: $\sin(2x) = 0$

   The solutions for $\sin(2x) = 0$ on $[0, \pi]$ are: $2x = 0, \pi, 2\pi$ which gives: $x = 0, \frac{\pi}{2}, \pi$

Step 2: Evaluate $f(x)$ at the critical points and endpoints

Now we calculate $f(x)$ at $x = 0$, $x = \frac{\pi}{2}$, and $x = \pi$ to determine the absolute minimum and maximum values.

1. At $x = 0$: $f(0) = 5 \cos^2(0) = 5 \cdot 1 = 5$

2. At $x = \frac{\pi}{2}$: $f\left(\frac{\pi}{2}\right) = 5 \cos^2\left(\frac{\pi}{2}\right) = 5 \cdot 0 = 0$

3. At $x = \pi$: $f(\pi) = 5 \cos^2(\pi) = 5 \cdot 1 = 5$

Step 3: Determine the absolute extreme values

From the evaluations above:

• The absolute maximum value of $f(x)$ on $[0, \pi]$ is $5$, occurring at $x = 0$ and $x = \pi$.
• The absolute minimum value of $f(x)$ on $[0, \pi]$ is $0$, occurring at $x = \frac{\pi}{2}$.

Final Answer

• Absolute Maximum: $f(x) = 5$ at $x = 0$ and $x = \pi$.
• Absolute Minimum: $f(x) = 0$ at $x = \frac{\pi}{2}$.

Would you like more details on any step, or do you have any questions?

Here are some related questions to consider:

1. How do we use trigonometric identities to simplify derivatives?
2. Why is it necessary to evaluate endpoints when finding absolute extrema on a closed interval?
3. How would the answer change if the interval were $(0, \pi)$ instead of $[0, \pi]$?
4. What are the double-angle identities, and why are they useful in trigonometry?
5. How can we verify the nature (maximum or minimum) of a critical point?

Tip: Always consider the interval boundaries when looking for absolute extrema on a closed interval; they can sometimes yield extreme values not found by only checking critical points.