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To find the critical points of the function \( f(x) = \sin(7x) \cos(7x) \), we first need to find the derivative of \( f(x) \) and set it equal to zero.

### Step 1: Rewrite the Function
We can simplify the function using a trigonometric identity:
\[
\sin(7x) \cos(7x) = \frac{1}{2} \sin(14x)
\]
Thus, we can rewrite \( f(x) \) as:
\[
f(x) = \frac{1}{2} \sin(14x)
\]

### Step 2: Find \( f'(x) \)
Now we differentiate \( f(x) \) with respect to \( x \):
\[
f'(x) = \frac{1}{2} \cdot 14 \cos(14x) = 7 \cos(14x)
\]

### Step 3: Set \( f'(x) = 0 \) to Find Critical Points
To find the critical points, we set \( f'(x) = 0 \):
\[
7 \cos(14x) = 0
\]
Dividing both sides by 7:
\[
\cos(14x) = 0
\]

### Step 4: Solve for \( x \)
The cosine function equals zero at \( 14x = \frac{\pi}{2} + n\pi \) for integer \( n \). Therefore:
\[
x = \frac{\pi}{28} + \frac{n\pi}{14}
\]

### Summary of Critical Points
The critical points of \( f(x) = \sin(7x) \cos(7x) \) are:
\[
x = \frac{\pi}{28} + \frac{n\pi}{14}, \quad n \in \mathbb{Z}
\]

Would you like further details on any part of this process, or have any questions? 

Here are 5 related questions for further exploration:
1. How do we determine if each critical point is a maximum, minimum, or saddle point?
2. What is the second derivative of \( f(x) = \sin(7x) \cos(7x) \)?
3. How would the critical points change if the function were \( f(x) = \sin(3x) \cos(3x) \)?
4. Can we use other methods (such as graphing) to confirm these critical points?
5. What are the key differences in behavior between trigonometric functions and polynomial functions in critical point analysis?

**Tip:** When dealing with trigonometric functions, consider using identities to simplify expressions before differentiating.

Find the critical points of the function f(x) = sin(7x)cos(7x).

Discover how to find the critical points of the function f(x) = sin(7x)cos(7x) by using trigonometric identities and calculus. This solution covers key steps including simplifying trigonometric expressions, calculating derivatives, and solving for zeros. Ideal for students in high school calculus or introductory college calculus courses.

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