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We are tasked with analyzing the function \( f(x) = \log(\cos(x)) \) to determine its behavior on two intervals:

1. \( \left( \frac{\pi}{2}, \pi \right) \): Show that \( f(x) \) is strictly increasing.
2. \( \left( 0, \frac{\pi}{2} \right) \): Show that \( f(x) \) is strictly decreasing.

### Solution:
To determine whether \( f(x) = \log(\cos(x)) \) is increasing or decreasing, we need to compute its derivative \( f'(x) \).

#### Step 1: Compute the derivative of \( f(x) \)
We apply the chain rule. The derivative of \( \log(\cos(x)) \) is:

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

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

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

- \( f'(x) = -\tan(x) \), so the behavior of the function depends on the sign of \( \tan(x) \).
  
##### (i) Interval \( \left( \frac{\pi}{2}, \pi \right) \)
- In this interval, \( \tan(x) \) is negative because both sine and cosine are negative.
- Since \( \tan(x) < 0 \), we have \( f'(x) = -\tan(x) > 0 \).
- Therefore, \( f(x) \) is strictly increasing on \( \left( \frac{\pi}{2}, \pi \right) \).

##### (ii) Interval \( \left( 0, \frac{\pi}{2} \right) \)
- In this interval, \( \tan(x) \) is positive because both sine and cosine are positive.
- Since \( \tan(x) > 0 \), we have \( f'(x) = -\tan(x) < 0 \).
- Therefore, \( f(x) \) is strictly decreasing on \( \left( 0, \frac{\pi}{2} \right) \).

### Conclusion:
- \( f(x) \) is strictly increasing on \( \left( \frac{\pi}{2}, \pi \right) \).
- \( f(x) \) is strictly decreasing on \( \left( 0, \frac{\pi}{2} \right) \).

Would you like further details on any part of this explanation?

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Here are 5 related questions for further practice:

1. What is the domain of the function \( \log(\cos(x)) \)?
2. How does the behavior of \( f(x) \) change if we consider \( \log(\sin(x)) \) instead?
3. How would you determine the concavity of \( \log(\cos(x)) \)?
4. What is the behavior of \( f(x) = \log(\cos(x)) \) as \( x \to 0 \)?
5. What would be the behavior of the function \( \log(\tan(x)) \) on the interval \( (0, \pi/2) \)?

**Tip**: Always check the sign of the derivative to determine the increasing or decreasing nature of a function over an interval.

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

Explore the behavior of f(x) = log(cos(x)) and show that it is strictly increasing in the interval (π/2, π) and strictly decreasing in the interval (0, π/2). This solution involves calculus concepts such as the derivative and monotonicity.

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