Hey guys! Ever wondered where a trig function like sin(3x)cos(3x) is heading – is it going up, or is it going down? That's what we're diving into today! We'll be using calculus to figure out the intervals of increase and decrease for this function. This is super useful for understanding the behavior of trigonometric functions, and it's a fundamental concept in calculus. So, let's get started and break down how to determine where sin(3x)cos(3x) is increasing or decreasing. This knowledge is important for graphing the function, determining its maximum and minimum values, and understanding its overall behavior. It helps in various fields, from physics and engineering to signal processing and data analysis, where understanding the behavior of oscillating functions is crucial. Plus, it’s just plain cool to see how math reveals the patterns in these functions! Let's get started by simplifying our lives a bit! Remember the double-angle identity? It's our best friend here. We can rewrite sin(3x)cos(3x) using the identity: sin(2θ) = 2sin(θ)cos(θ). This simplifies our function significantly and makes the calculus a lot easier! We'll show you how to do it in an easy to understand way so you guys get it. Because, let’s be honest, complex math can be a bit overwhelming, but when you break it down, it's totally manageable. Ready? Let's roll!

Simplifying the Function Using Trigonometric Identities

Alright, let's start with the basics, shall we? Before we dive into finding where our function increases and decreases, let's simplify it. Our function is sin(3x)cos(3x). Now, this looks a bit messy, but there's a handy trigonometric identity that will make our lives much easier. Remember the double-angle identity: sin(2θ) = 2sin(θ)cos(θ)? Perfect! We can manipulate our function to fit this identity. We will have to make some changes to match with the formula, so we can see how simple this becomes. To make it work, we can multiply and divide our function by 2: sin(3x)cos(3x) = (1/2) * 2sin(3x)cos(3x). See that 2sin(3x)cos(3x) part? It's practically screaming to be replaced with sin(2 * 3x), which simplifies to sin(6x). So, now our function becomes: (1/2)sin(6x). Isn't that much cleaner? This rewritten form, (1/2)sin(6x), is the one we will be using for the remainder of this analysis. It's much simpler to differentiate and analyze. Using trigonometric identities not only simplifies calculations but also reveals the underlying structure and properties of the function. Now we can proceed with confidence, knowing we're working with a more manageable form. Always remember that simplification is a powerful tool in mathematics! Simplification makes the next steps, like finding the derivative and determining increasing and decreasing intervals, way more straightforward. This is a common strategy in calculus, helping you deal with complex expressions by transforming them into simpler, more manageable forms. This will also give us the chance to see how we'll solve other complicated trigonometric function problems.

Finding the Derivative of the Simplified Function

Now that we have our simplified function, (1/2)sin(6x), the next step is to find its derivative. The derivative tells us about the rate of change of the function – in other words, whether it's increasing or decreasing at any given point. To find the derivative, we'll use the chain rule. Remember the chain rule? It states that if we have a composite function, like our sin(6x), we take the derivative of the outer function (sine) and multiply it by the derivative of the inner function (6x). So, the derivative of sin(6x) is cos(6x) multiplied by the derivative of 6x, which is 6. Therefore, the derivative of (1/2)sin(6x) is: (1/2) * 6cos(6x) = 3cos(6x). That's it! Easy, right? We've successfully found the derivative of our function. The derivative, 3cos(6x), is the key to understanding where the original function is increasing or decreasing. This is because the sign of the derivative tells us the direction of the function's slope. A positive derivative means the function is increasing, and a negative derivative means the function is decreasing. The derivative gives us the slope of the original function at any given point. A positive slope indicates the function is increasing, while a negative slope indicates it is decreasing. This concept is fundamental to understanding the behavior of functions. Finding the derivative is the cornerstone of determining the intervals of increase and decrease. The derivative provides us with the necessary information to analyze the function's slope and, consequently, its behavior. From here, the next steps will be easy, and we will get to see a complete picture of the behavior of our function!

Determining Critical Points and Intervals

Now that we have our derivative, 3cos(6x), we need to find the critical points. Critical points are the points where the derivative is either equal to zero or undefined. These are the potential points where the function changes direction. Let's find them: Setting the derivative equal to zero: 3cos(6x) = 0. Divide both sides by 3: cos(6x) = 0. We know that cosine is zero at odd multiples of π/2. So, we can write: 6x = (2n + 1)π/2, where n is an integer. Now, solve for x: x = (2n + 1)π/12. These are our critical points! Because the cosine function is smooth and continuous, there are no points where the derivative is undefined. Now, let's determine the intervals of increase and decrease. We'll divide the real number line into intervals using our critical points: x = (2n + 1)π/12. Since the cosine function is continuous, we don't have to worry about points where the derivative doesn't exist. To determine whether the function is increasing or decreasing in each interval, we can use a test point within each interval and plug it into the derivative, 3cos(6x). If the result is positive, the function is increasing; if the result is negative, the function is decreasing. The critical points are the dividing lines between the intervals of increase and decrease. The function's behavior changes at these points, making them crucial for analysis. To determine the direction of the function, we test the sign of the derivative in each interval, allowing us to accurately describe where the function rises and falls. Finding these critical points is a crucial step in understanding the function's behavior, and it sets the stage for the final determination of the intervals of increase and decrease. Now, let's test these intervals!

Testing Intervals of Increase and Decrease

Alright, now it's time to get our hands dirty and test some intervals! We'll pick test points between our critical points, x = (2n + 1)π/12, and plug them into our derivative, 3cos(6x). Let's test a few intervals to understand the pattern. Remember, we are testing the derivative, 3cos(6x), because the sign of the derivative tells us whether the original function is increasing or decreasing. Let’s start with an easy one. When n=0, x = π/12, the critical point is π/12. We get intervals of (-∞, π/12), (π/12, 3π/12), (3π/12, 5π/12) and so on. Let's pick a test point in the interval (0, π/12). Let’s try x = π/24. Plugging it into the derivative: 3cos(6 * π/24) = 3cos(π/4). Since cos(π/4) is positive, the derivative is positive, meaning the function is increasing in this interval. Now, let's pick a test point in the interval (π/12, 3π/12), say x = π/8. Plugging it into the derivative: 3cos(6 * π/8) = 3cos(3π/4). Since cos(3π/4) is negative, the derivative is negative, and the function is decreasing in this interval. Continuing this pattern, we can test other intervals. Let's pick a test point in the interval (3π/12, 5π/12), say x = π/3. Plugging it into the derivative: 3cos(6 * π/3) = 3cos(2π). Since cos(2π) is positive, the derivative is positive, and the function is increasing in this interval. You will start to see a pattern emerging. The function alternates between increasing and decreasing intervals. We can continue this process for all the intervals. This process is key to fully understanding the function's behavior. The results help us to understand where the function's value increases and decreases in a continuous manner. Doing so, we're building a clear picture of how the function behaves across its entire domain, giving us a complete understanding of its behavior. Knowing the sign of the derivative allows us to precisely identify where the original function is rising or falling, providing a comprehensive understanding of its behavior. You can use a table to record these results for better visualization, as this helps to clarify the function's overall behavior. So, let’s look at a summary of the intervals.

Summary of Intervals of Increase and Decrease

Okay, let's wrap this up with a nice, clean summary of what we've found! We've done the hard work, so here’s a simplified version. Based on our calculations, the function (1/2)sin(6x) – which is the same as sin(3x)cos(3x) – increases and decreases in intervals determined by the critical points x = (2n + 1)π/12, where n is an integer. Here's a general description: The function increases on the intervals where the derivative is positive. The function decreases on the intervals where the derivative is negative. The function increases on intervals like ((2n)π/12, ((2n+1)π)/12) . The function decreases on intervals like (((2n+1)π)/12, ((2n+2)π)/12) For a more detailed look, the intervals are: The function increases on (0, π/12), (π/2, 5π/12), (π, 13π/12), ... The function decreases on (π/12, 3π/12), (5π/12, 7π/12), (13π/12, 15π/12), ... This means the function goes up, then down, then up again, creating a wave-like pattern. This behavior is typical of trigonometric functions. Understanding these intervals is crucial for sketching the graph of sin(3x)cos(3x), and for understanding its behavior in more complex mathematical problems. This also helps visualize the function's graph and to identify points of maxima and minima. The analysis gives us a complete picture of the function's behavior, showing us exactly where it's climbing and where it's falling. We have now fully analyzed the function, finding its increasing and decreasing intervals! Isn’t this awesome? We've successfully determined the intervals of increase and decrease for sin(3x)cos(3x). Pretty cool, right? You've learned how to simplify, differentiate, find critical points, and analyze intervals using the derivative. This is a fundamental concept in calculus, and you've got it down! Keep practicing, and you'll become a pro at analyzing the behavior of functions. So keep learning and don’t give up. The more you know, the more confident you'll feel when tackling tougher math problems. Happy calculating, and keep exploring the amazing world of math! And remember guys, every mathematical concept builds on previous ones, so keep building your knowledge. Well done, guys! You did great!