Sin3x Cos3x: Unveiling Increasing And Decreasing Intervals

Hey math enthusiasts! Today, we're diving into the fascinating world of trigonometry and calculus, specifically focusing on the function sin(3x)cos(3x). Our goal? To understand where this function is increasing and where it's decreasing. This is a classic problem that combines trigonometric identities with the power of derivatives, providing a fantastic exercise in mathematical analysis. So, grab your pencils and let's get started. This article breaks down the steps to find the intervals and determine the increasing and decreasing regions of the function. We will also touch upon the related concepts to better understand it.

Understanding the Basics: Trigonometric Identities and Derivatives

Before we jump into the core of the problem, let's brush up on some essential concepts. We'll need to use trigonometric identities to simplify the function, and then we'll employ derivatives to analyze its behavior. Understanding these elements is absolutely crucial before we begin. Are you ready?

First, a trigonometric identity will be helpful: the double-angle formula for sine. Remember the identity sin(2θ) = 2sin(θ)cos(θ)? Well, it's a game-changer here. Our function, sin(3x)cos(3x), looks similar to the right side of this identity. We can rewrite it using this trick. Multiply and divide the function by 2. This will help us use the trigonometric identity and simply the function. Let’s do it step by step:

f(x) = sin(3x)cos(3x)
f(x) = (1/2) * 2sin(3x)cos(3x)
f(x) = (1/2)sin(6x)

See? Using the identity simplifies things beautifully, making it easier to work with. Now we have f(x) = (1/2)sin(6x). This simplifies things significantly. This will help with the derivative process.

Next, we need to talk about derivatives. The derivative of a function tells us its rate of change. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. The derivative of sin(ax) is acos(ax), where 'a' is a constant. We will use this in the next sections. Stay tuned!

Calculating the Derivative of sin(3x)cos(3x)

Alright, it's time to find the derivative of our simplified function, f(x) = (1/2)sin(6x). This is where calculus comes into play. We're going to use the chain rule here, because the 6x inside the sine function is not simply x. The chain rule states that if you have a function within a function (a composite function), you take the derivative of the outer function, keeping the inner function the same, and then multiply it by the derivative of the inner function. If this seems confusing, don't worry, we'll break it down.

So, the derivative of sin(6x) is 6cos(6x). Applying the derivative to our function:

f'(x) = (1/2) * 6cos(6x)
f'(x) = 3cos(6x)

Therefore, the derivative of f(x) = sin(3x)cos(3x) is f'(x) = 3cos(6x). We now have our first derivative which is f'(x) = 3cos(6x). This is super important because it provides a map of the function's rate of change across the x-axis. As a reminder: If f'(x) > 0, then the function f(x) is increasing. If f'(x) < 0, then the function f(x) is decreasing. Using this concept we will be able to determine in which intervals is increasing and decreasing.

Finding Critical Points

Now, let's find the critical points of the function. Critical points are the points where the derivative is either equal to zero or undefined. These points are significant because they often mark the transition points where the function changes from increasing to decreasing, or vice versa. The critical points will help determine the interval in which the function is increasing or decreasing.

In our case, the derivative f'(x) = 3cos(6x) is defined everywhere. We only need to find where it equals zero.

3cos(6x) = 0
cos(6x) = 0

The cosine function is zero at odd multiples of π/2. So, we can write:

6x = (2n + 1)π/2, where n is an integer.
x = (2n + 1)π/12

So, the critical points are x = (2n + 1)π/12. These are the points where the function might change from increasing to decreasing (or vice versa). These are extremely important in determining the increasing and decreasing intervals. Got it?

Determining the Intervals of Increase and Decrease

Alright, let's put it all together. We have our critical points: x = (2n + 1)π/12. We need to use these critical points to divide the x-axis into intervals. We then test the sign of the derivative, f'(x) = 3cos(6x), within each interval to determine whether the function is increasing or decreasing. Now, remember the derivative tells us about the slope of the function at any point. A positive slope indicates the function is increasing, and a negative slope indicates the function is decreasing. It is important to remember that we are working with radians.

Let's consider the interval between 0 and π/12. For simplicity let's consider the interval when n = 0. So, x = π/12. If we test a value in this interval, such as x = π/24, we get:

f'(π/24) = 3cos(6 * π/24)
f'(π/24) = 3cos(π/4)
f'(π/24) = 3 * (√2/2)
f'(π/24) > 0

Since the derivative is positive, the function is increasing in the interval (0, π/12). Then, consider the interval (π/12, 3π/12). Test a value such as x = π/6:

f'(π/6) = 3cos(6 * π/6)
f'(π/6) = 3cos(π)
f'(π/6) = -3
f'(π/6) < 0

Since the derivative is negative, the function is decreasing in the interval (π/12, 3π/12). We can continue this process for other intervals as well, you'll observe the function alternates between increasing and decreasing. Specifically, the function is increasing in the intervals ((2n)π/12, ((2n+1)π/12)) and decreasing in the intervals (((2n+1)π/12), ((2n+2)π/12)), where n is an integer.

Visualizing with a Graph

Visualizing the function f(x) = sin(3x)cos(3x) on a graph will solidify your understanding. The graph will show you the ups and downs, the peaks and valleys, and the points where the function changes direction. You'll clearly see the intervals where the function is increasing and decreasing, as we've calculated. Plotting a graph is always useful, since this gives you a visual proof and provides a good understanding of the interval of increasing and decreasing.

When you graph f(x) = (1/2)sin(6x), you will see a wave-like pattern, oscillating between -1/2 and 1/2. The function completes six cycles in the interval of 2π. The graph confirms that our calculated intervals of increase and decrease are correct.

Practical Implications and Applications

Understanding the increasing and decreasing behavior of a function has practical applications. It is useful in many real-world scenarios, from optimizing a product's output to understanding the changing patterns. This can be used in physics, engineering, and economics.

For example, in physics, the function might represent the displacement of an oscillating object. Understanding the intervals where the displacement is increasing or decreasing helps analyze the object's motion. In engineering, analyzing such functions can be part of designing systems to optimize performance. In economics, it might be used to analyze stock prices. Therefore, the more we understand the behavior of the functions the more we understand the real-world applications.

Conclusion: Mastering the Art of Analysis

And there you have it! We've successfully analyzed the increasing and decreasing intervals of the function sin(3x)cos(3x). We used trigonometric identities, derivatives, critical points, and interval testing to come to this conclusion. This process provides a powerful example of how calculus can be used to understand the behavior of functions. Remember that the journey of understanding is not always easy, but it is always rewarding. The more you practice, the more intuitive the process becomes. Keep experimenting, keep exploring, and you'll find that the world of mathematics is full of exciting discoveries! So, keep up the fantastic work, and happy calculating!

This method can be applied to other trigonometric functions and is useful for many other problems. Keep practicing and keep up the great work!