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To find the critical points of the function \( f(x) = \sin(5x) \cos(5x) \), we need to take its derivative and set it equal to zero.

### Step 1: Simplify \( f(x) \)
Using the double-angle identity for sine:
\[
\sin(5x) \cos(5x) = \frac{1}{2} \sin(10x)
\]
Thus, we can rewrite \( f(x) \) as:
\[
f(x) = \frac{1}{2} \sin(10x)
\]

### Step 2: Differentiate \( f(x) \)
Now, let’s find \( f'(x) \):
\[
f'(x) = \frac{1}{2} \cdot 10 \cos(10x) = 5 \cos(10x)
\]

### Step 3: Set \( f'(x) = 0 \)
To find the critical points, we set the derivative equal to zero:
\[
5 \cos(10x) = 0
\]
\[
\cos(10x) = 0
\]

### Step 4: Solve \( \cos(10x) = 0 \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):
\[
10x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}
\]
Solving for \( x \), we get:
\[
x = \frac{\pi}{20} + \frac{k \pi}{10}
\]

### Step 5: Match the Answer
The critical points are given by:
\[
x = \frac{\pi}{20} + k \frac{\pi}{10}
\]
This corresponds to **Option A**:
\[
\text{The critical point(s) occur at } x = \frac{\pi}{20} + k \cdot \frac{\pi}{10}.
\]

### Final Answer
The correct answer is:
**A. The critical point(s) occur at \( x = \frac{\pi}{20} + k \cdot \frac{\pi}{10} \)**.

Would you like further explanation or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do we determine critical points for functions that don't simplify easily?
2. What is the significance of the double-angle identity in trigonometry?
3. How do you verify if a critical point is a maximum, minimum, or saddle point?
4. What are the possible values of \( k \) in this context, and how does it affect the set of critical points?
5. How would the process change if \( f(x) \) had more complex trigonometric components?

**Tip:** Remember that trigonometric identities, like the double-angle formula, can simplify differentiation tasks significantly by reducing complex expressions.

Find the critical points of the following function: f(x) = sin(5x) cos(5x). Let k be an integer. Select the correct choice and, if necessary, fill in the answer box to complete your choice.

Learn how to find the critical points of the function f(x) = sin(5x) cos(5x) using calculus and trigonometry. This problem uses the double-angle identity and involves setting the derivative equal to zero. Suitable for students in Grades 11-12.

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