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To find the \( x \) values where the tangent line is horizontal for the function:

\[
f(x) = -3 \sin x \cos x
\]

we need to follow these steps:

### 1. Find the derivative \( f'(x) \)
To determine where the tangent line is horizontal, we need to find where the slope of the tangent line (the derivative) is equal to zero. So, we first calculate the derivative of \( f(x) \).

Using the product rule:
\[
f(x) = -3 \sin x \cos x
\]

The product rule states that if \( f(x) = u(x) \cdot v(x) \), then:
\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]

Here, let \( u(x) = \sin x \) and \( v(x) = \cos x \). Applying the product rule:

\[
f'(x) = -3 [(\cos x)(\cos x) + (\sin x)(-\sin x)]
\]

Simplifying:
\[
f'(x) = -3 [\cos^2 x - \sin^2 x]
\]

We can recognize this as the double-angle identity:
\[
f'(x) = -3 \cos(2x)
\]

### 2. Set the derivative equal to zero
To find where the tangent line is horizontal, we set the derivative equal to zero:
\[
-3 \cos(2x) = 0
\]

This simplifies to:
\[
\cos(2x) = 0
\]

### 3. Solve for \( x \)
The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \), so:
\[
2x = \frac{\pi}{2} + n\pi \quad \text{where} \, n \, \text{is an integer}.
\]

Solving for \( x \):
\[
x = \frac{\pi}{4} + \frac{n\pi}{2}
\]

Thus, the values of \( x \) where the tangent line is horizontal are:

\[
x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \dots
\]

This represents all solutions.

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Would you like more detailed steps or have any questions about this solution? Here are some related questions:

1. What does the double-angle identity for cosine represent in trigonometry?
2. Can you explain how the product rule is applied to find derivatives?
3. How do the horizontal tangent lines relate to critical points on the graph of a function?
4. How can we verify that the solutions we found are correct by graphing the function?
5. What other types of problems use the concept of horizontal tangents?

**Tip:** For trigonometric functions, recognizing standard identities like the double-angle formula can simplify finding derivatives and solving equations.

Find all x values on the graph of f(x) = -3 sin(x) cos(x) where the tangent line is horizontal.

Discover how to find x values where the tangent line is horizontal for the function f(x) = -3 sin(x) cos(x). This solution covers calculus concepts like derivatives and uses trigonometric identities. Suitable for students in Grades 10-12.

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