Hey there, math enthusiasts! Ever wondered about the ups and downs of trigonometric functions? Today, we're diving deep into the world of trigonometry to explore whether the function sin(3x)cos(3x) is increasing or decreasing. This might sound a bit complex, but trust me, we'll break it down step by step, making it easy to understand. We'll explore the concepts of increasing and decreasing functions, and then apply these ideas to sin(3x)cos(3x). Buckle up, because we're about to embark on an exciting mathematical journey!

Understanding Increasing and Decreasing Functions

Before we jump into sin(3x)cos(3x), let's get our fundamentals straight. What exactly does it mean for a function to be increasing or decreasing? In simple terms, a function is increasing if its output (the y-value) gets larger as the input (the x-value) increases. Think of it like climbing a hill: as you move to the right (increase x), you also go up (increase y). Mathematically, if f(x₁) < f(x₂) for all x₁ < x₂, the function f(x) is increasing.

On the flip side, a function is decreasing if its output gets smaller as the input increases. This is like going downhill: as you move to the right (increase x), you go down (decrease y). Mathematically, if f(x₁) > f(x₂) for all x₁ < x₂, the function f(x) is decreasing. It is important to remember that these definitions are about the overall trend of a function within a certain interval. Functions can be increasing over some intervals and decreasing over others. Further, a function can also remain constant, or neither increase nor decrease, within an interval.

To determine whether a function is increasing or decreasing, calculus provides powerful tools like the first derivative test. The first derivative, f'(x), tells us the slope of the tangent line to the function at any given point. If f'(x) > 0, the function is increasing at that point; if f'(x) < 0, the function is decreasing at that point; and if f'(x) = 0, the function is neither increasing nor decreasing at that point (it might be a constant, a local maximum, or a local minimum). This is the key to understanding the behavior of sin(3x)cos(3x). Let’s get into the nitty-gritty of this function to determine what is happening.

Transforming sin(3x)cos(3x) for Easier Analysis

Alright guys, let's talk about our function, sin(3x)cos(3x). At first glance, it might seem a bit complicated. However, we can use a clever trick to simplify it using a trigonometric identity. Specifically, we can use the double-angle identity for sine, which states that sin(2θ) = 2sin(θ)cos(θ). Now, if we let θ = 3x, then 2θ = 6x. This suggests that we can rewrite our original function. But first, let’s quickly manipulate the original function to look similar to the double-angle identity. To do this, we need to multiply and divide by 2. Thus, the original function can be written as (1/2) * 2sin(3x)cos(3x). Notice that the expression, 2sin(3x)cos(3x), is very similar to the right-hand side of the double-angle identity. Thus, we have (1/2) * sin(6x).

So, our function sin(3x)cos(3x) is equivalent to (1/2)sin(6x). This is a much easier function to analyze! We now need to determine if this new function is increasing or decreasing. Now, you may ask how this transformation is useful. Using this form simplifies the process of finding the derivative, and we can directly apply the derivative rules to (1/2)sin(6x). Now, we’re dealing with a simple sine function, which we can easily analyze for its increasing and decreasing behavior.

Finding the Derivative of (1/2)sin(6x)

Okay, team, now that we've simplified our function to (1/2)sin(6x), the next step is to find its derivative. The derivative will tell us the slope of the function at any point, which is crucial for determining if it’s increasing or decreasing. To find the derivative, we'll use the chain rule. Remember, the chain rule is used when differentiating composite functions (functions within functions). In our case, we have a function within a function - sin(6x), where 6x is a function of x.

The derivative of sin(x) is cos(x). Using the chain rule, the derivative of sin(6x) is cos(6x) * 6. Then, we must include the constant, 1/2. Thus, the derivative of (1/2)sin(6x) is (1/2) * cos(6x) * 6, which simplifies to 3cos(6x). So, the first derivative of our original function is 3cos(6x). This 3cos(6x) is the slope of the tangent line at any point x. Using this derivative, we can figure out where the function is increasing or decreasing. Let’s see how!

Analyzing the Derivative: Increasing or Decreasing Intervals

Now for the big question: how do we use the derivative 3cos(6x) to figure out where (1/2)sin(6x) is increasing or decreasing? Remember what we talked about earlier: f'(x) > 0 means the function is increasing, and f'(x) < 0 means the function is decreasing. The function cos(6x) oscillates between -1 and 1.

Specifically, cos(6x) = 1 when 6x = 0, 2π, 4π…, or when x = 0, π/3, 2π/3… In these intervals, f'(x) = 3cos(6x) > 0, meaning that the function is increasing. Also, cos(6x) = -1 when 6x = π, 3π, 5π…, or when x = π/6, π/2, 5π/6… In these intervals, f'(x) = 3cos(6x) < 0, meaning that the function is decreasing. Since cosine oscillates, our function, (1/2)sin(6x), is neither always increasing nor always decreasing. The function oscillates.

So, sin(3x)cos(3x) (or its equivalent, (1/2)sin(6x)) is increasing on the intervals where 3cos(6x) > 0, and it's decreasing on the intervals where 3cos(6x) < 0. These intervals occur because of the nature of the cosine function. The function will be increasing when the cosine function is positive, and decreasing when the cosine function is negative. The original function does not always increase or decrease, but oscillates, reflecting the cyclical behavior of the trigonometric functions. The function shows a pattern of increase and decrease. The rate of increase and decrease also varies, based on the derivative.

Conclusion: The Ups and Downs of sin(3x)cos(3x)

Alright, folks, we've come to the end of our trigonometric adventure! We started with sin(3x)cos(3x), simplified it using a trigonometric identity to (1/2)sin(6x), found its derivative 3cos(6x), and analyzed the derivative to determine its increasing and decreasing intervals. We've discovered that sin(3x)cos(3x) doesn’t just increase or just decrease; it does both! The function increases and decreases periodically due to the cyclical nature of the sine and cosine functions. It's a great example of how understanding derivatives can help us analyze the behavior of functions. This is a common phenomenon in the field of signal processing. Thus, analyzing and understanding these principles can help you grasp more advanced concepts.

So, the next time you encounter sin(3x)cos(3x), you'll know exactly what's happening: it's a function that's constantly changing, oscillating between increasing and decreasing behaviors. Keep practicing, keep exploring, and keep the math fun! Until next time, happy calculating, and keep those derivatives flowing!