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To solve the problem, we need to analyze the behavior of the function \( f(x) = \log(\cos(x)) \) in the given intervals.

### Step 1: Find the derivative of \( f(x) \)
The derivative of \( f(x) = \log(\cos(x)) \) can be found using the chain rule. Let’s differentiate:

\[
f'(x) = \frac{d}{dx} \left( \log(\cos(x)) \right) = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)
\]

So, \( f'(x) = -\tan(x) \).

### Step 2: Analyze the sign of \( f'(x) \)

- **(i) On the interval \( \left(\frac{\pi}{2}, \pi \right) \):**
  - In this interval, the tangent function \( \tan(x) \) is **negative** because \( \sin(x) \) is positive and \( \cos(x) \) is negative. Hence, \( f'(x) = -\tan(x) \) will be **positive**, implying that \( f(x) \) is **increasing** in this interval.

- **(ii) On the interval \( \left(0, \frac{\pi}{2}\right) \):**
  - In this interval, the tangent function \( \tan(x) \) is **positive** because both \( \sin(x) \) and \( \cos(x) \) are positive. Therefore, \( f'(x) = -\tan(x) \) will be **negative**, implying that \( f(x) \) is **decreasing** in this interval.

### Conclusion:
- \( f(x) = \log(\cos(x)) \) is **strictly increasing** in \( \left( \frac{\pi}{2}, \pi \right) \).
- \( f(x) = \log(\cos(x)) \) is **strictly decreasing** in \( \left( 0, \frac{\pi}{2} \right) \).

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

### Follow-up questions:
1. What is the significance of a strictly increasing or decreasing function?
2. Can you explain how the tangent function behaves in different intervals?
3. How does the domain of \( \log(\cos(x)) \) affect the analysis?
4. Why does the function become undefined at \( x = \frac{\pi}{2} \)?
5. What other methods could you use to analyze the behavior of \( f(x) \)?

### Tip:
When working with trigonometric functions, always be mindful of their periodic nature and undefined points (such as when \( \cos(x) = 0 \), where \( f(x) \) may not exist).

Show that f(x) = log(cos x) is (i) strictly increasing in (π/2, π) (ii) strictly decreasing in (0, π/2).

In this problem, learn how to analyze the monotonic behavior of f(x) = log(cos x). This includes finding the derivative, understanding the behavior of tan(x), and determining where the function is increasing and decreasing in the intervals (π/2, π) and (0, π/2), respectively. Suitable for students in Grades 11-12.

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