Hey there, math enthusiasts! Today, we're diving into the fascinating world of trigonometry and calculus to figure out where the function f(x) = sin(3x)cos(3x) is increasing or decreasing. Sounds fun, right? Don't worry, we'll break it down step by step, so even if you're not a math whiz, you'll still be able to follow along. This journey involves understanding the behavior of trigonometric functions, using some calculus magic (derivatives!), and visualizing the function's ups and downs. Buckle up; it's going to be a fun ride!

Unveiling the Basics: Trigonometry and Functions

Alright, before we get our hands dirty with the sin(3x)cos(3x) function, let's refresh our memories on some fundamental concepts. First off, what exactly is a trigonometric function? Simply put, it's a function that relates the angles of a triangle to the lengths of its sides. The most common ones are sine (sin), cosine (cos), and tangent (tan). These functions are periodic, meaning their values repeat over a specific interval. For instance, the sine and cosine functions oscillate between -1 and 1, completing a full cycle every 2π radians (or 360 degrees). Understanding this periodic nature is crucial because it directly influences how our function sin(3x)cos(3x) behaves. The argument inside the trigonometric functions, like the '3x' in our case, affects the function's period and frequency. A larger coefficient (like '3') will compress the graph horizontally, leading to more cycles within a given interval. So, keep this in mind: our function sin(3x)cos(3x) will complete three cycles within the same interval that a simple sin(x) or cos(x) function would complete only one.

Now, what about the term 'function'? In mathematical terms, a function is a rule that assigns each input value (x) to exactly one output value (y or f(x)). In our case, the input is an angle x, and the output is the value of sin(3x)cos(3x) for that angle. Graphing a function helps us visualize its behavior. We can see where it increases (goes up), decreases (goes down), and reaches its maximum and minimum values. This is where calculus, particularly derivatives, comes into play. The derivative of a function tells us its rate of change at any point. A positive derivative means the function is increasing, a negative derivative means it's decreasing, and a derivative of zero indicates a critical point (a potential maximum or minimum). That's a lot of things, right? But don't you worry, the important thing is that you understand the basics.

The Role of Derivatives

To figure out the increasing and decreasing intervals, we'll use the power of calculus, specifically derivatives. The derivative of a function, denoted as f'(x), tells us the rate of change of the function at any point. When f'(x) > 0, the function is increasing; when f'(x) < 0, the function is decreasing; and when f'(x) = 0, we have a critical point (potentially a maximum or minimum). So, our mission is to find the derivative of f(x) = sin(3x)cos(3x), analyze its sign, and pinpoint the intervals where the function is either climbing or descending. This is like having a roadmap to understand the behavior of the function.

Getting Our Hands Dirty: Finding the Derivative

Okay, time to roll up our sleeves and calculate the derivative of f(x) = sin(3x)cos(3x). This requires using the product rule and the chain rule. The product rule states that the derivative of a product of two functions, u(x) and v(x), is given by: (uv)' = u'v + uv'. The chain rule helps us differentiate composite functions (functions within functions). Let's see how it unfolds:

1. Rewrite the function: Before we start, let's use a trigonometric identity to simplify things. Recall that 2sin(A)cos(A) = sin(2A). Therefore, sin(3x)cos(3x) = 1/2 * sin(6x). This is much easier to work with!
2. Differentiate: Now, we'll find the derivative of f(x) = 1/2 * sin(6x). Using the chain rule, we have: f'(x) = 1/2 * cos(6x) * 6 = 3cos(6x).

Voila! The derivative of f(x) = sin(3x)cos(3x) is f'(x) = 3cos(6x). This is our key to unlocking the increasing and decreasing intervals.

Product Rule and Chain Rule Explained

Let's clarify how the product rule and the chain rule come into play. Initially, you might have been tempted to differentiate sin(3x)cos(3x) directly using the product rule. This is perfectly valid, but it requires a bit more work. Let's break it down:

• If we consider u(x) = sin(3x) and v(x) = cos(3x), then u'(x) = 3cos(3x) and v'(x) = -3sin(3x).
• Applying the product rule, (uv)' = u'v + uv' gives us: (sin(3x)cos(3x))' = (3cos(3x)cos(3x) + sin(3x)(-3sin(3x))) = 3cos²(3x) - 3sin²(3x).
• You could then use the double-angle identity for cosine, cos(2A) = cos²(A) - sin²(A), to simplify this to 3cos(6x). See, it's the same result, but more steps. The chain rule is essential for differentiating composite functions. When we differentiated sin(6x), we recognized that 6x is a function within the sine function. Thus, we first differentiated the outer function (sine), leaving the inner function (6x) as is, and then multiplied by the derivative of the inner function (which is 6). That is why we have cos(6x) * 6. Both rules are your best friends in calculus.

Pinpointing Increasing and Decreasing Intervals

Now that we have the derivative f'(x) = 3cos(6x), let's determine the intervals where the original function f(x) = sin(3x)cos(3x) is increasing or decreasing. Remember, the sign of the derivative tells us the function's behavior:

• Increasing: f'(x) > 0 (3cos(6x) > 0), which means cos(6x) > 0.
• Decreasing: f'(x) < 0 (3cos(6x) < 0), which means cos(6x) < 0.

To solve these inequalities, we need to analyze the cosine function. Cosine is positive in the first and fourth quadrants and negative in the second and third quadrants. However, since we have cos(6x), the period is compressed.

1. Find critical points: Set f'(x) = 0 and solve for x: 3cos(6x) = 0 => cos(6x) = 0. This occurs when 6x = π/2 + nπ, where n is an integer. Thus, x = π/12 + nπ/6. These are our critical points.
2. Test intervals: Divide the real number line into intervals using the critical points. For instance, consider the interval (0, π/12). Choose a test value, say x = π/24. Plug this into f'(x) = 3cos(6x): f'(π/24) = 3cos(6 * π/24) = 3cos(π/4) > 0. Since the derivative is positive, the function is increasing in this interval.
3. Generalize: Based on the periodic nature of the cosine function, we can generalize the intervals: The function f(x) = sin(3x)cos(3x) is:
   • Increasing in the intervals: (π/12 + nπ/6, π/6 + nπ/6), where n is an integer.
   • Decreasing in the intervals: (π/6 + nπ/6, π/12 + (n+1)π/6), where n is an integer.

Visualizing the Intervals

To make this clearer, let's imagine a number line, divided by these critical points. In each section, the sign of the derivative is the same, meaning our function consistently rises or falls. Where the derivative is positive, our original function climbs uphill. Where the derivative is negative, it goes downhill. The points at which it changes direction are where the derivative crosses zero, the critical points. This whole process helps us map out the ups and downs of f(x).

Conclusion: Mastering the Ups and Downs

And there you have it, folks! We've successfully navigated the ups and downs of f(x) = sin(3x)cos(3x). We used derivatives to determine the increasing and decreasing intervals, understanding how trigonometric identities and calculus work together. Keep in mind that math is all about practice. The more problems you solve, the more comfortable you'll become with these concepts. So, keep exploring, keep experimenting, and happy calculating!

Recap and Key Takeaways

• Simplify first: Always look for ways to simplify the function using trigonometric identities before differentiating. It saves time and reduces errors.
• Understand the rules: Master the product rule, the chain rule, and the properties of trigonometric functions. These are your essential tools.
• Visualize: Use graphs to help you visualize the behavior of the function and confirm your results. This gives you a quick and helpful check.
• Practice, practice, practice: The more problems you solve, the better you'll become at recognizing patterns and applying the correct techniques.

So next time you encounter a trigonometric function, remember the steps we've taken today. You're now equipped to determine where it increases and decreases, giving you a deeper understanding of its behavior. Keep exploring the world of math, and enjoy the journey!