Hey there, math enthusiasts! Ever wondered how to figure out where a trigonometric function like sin(3x)cos(3x) is going up or down? It's a classic calculus problem, and we're going to break it down together. Understanding the increasing and decreasing intervals of a function is super useful. It tells us about the function's behavior across its domain, highlighting where it's rising, falling, or staying flat. This knowledge is not only crucial for graphing the function accurately but also has implications in various fields like physics (analyzing wave behavior), engineering (understanding signal processing), and economics (modeling cyclical patterns). So, let's dive into how we can determine these intervals for our given function, sin(3x)cos(3x).

Unveiling the Function: sin(3x)cos(3x)

First things first, what exactly are we dealing with? The function f(x) = sin(3x)cos(3x) is a product of two trigonometric functions. At first glance, it might seem a bit complex, but there's a neat trick we can use to simplify things. Remember the double-angle identity for sine? It states that sin(2θ) = 2sin(θ)cos(θ). We can manipulate our function to fit this identity, which will make our calculations much easier. Recognize that sin(3x)cos(3x) is very similar to the right side of the double-angle identity. To make it exactly match, we'll need to multiply and divide by 2. Thus, we have:

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

Now, applying the double-angle identity:

f(x) = (1/2)sin(6x)

See how much cleaner that looks? This rewriting is crucial; it simplifies our differentiation and makes it much easier to analyze the function's behavior. We've transformed our original function into a much simpler form, which is essentially a scaled and horizontally compressed sine function. With this simplification, we can now move forward to find the critical points and determine where our function is increasing or decreasing.

Now that we've simplified our function to f(x) = (1/2)sin(6x), we're ready to proceed with finding its increasing and decreasing intervals. We'll utilize the power of calculus, specifically derivatives, to pinpoint where the function's slope is positive (increasing), negative (decreasing), or zero (at critical points). This process involves a few key steps: finding the derivative, determining critical points, and using a sign analysis. Let's get to work!

Finding the Derivative

To find the increasing and decreasing intervals, we need to know the function's rate of change, which is given by its derivative. The derivative of f(x) = (1/2)sin(6x) is found using the chain rule. Remember, the derivative of sin(u) is cos(u) * du/dx. Applying this, we get:

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

So, the derivative of f(x) is f'(x) = 3cos(6x). This derivative represents the slope of the function at any point x. A positive value of f'(x) indicates that the function is increasing at that point, while a negative value indicates that it's decreasing. The points where f'(x) = 0 or is undefined are the critical points, where the function's behavior might change.

Identifying Critical Points

Critical points are the heart of our analysis. These are the points where the function might switch from increasing to decreasing or vice versa. To find these, we set the derivative f'(x) = 3cos(6x) equal to zero and solve for x.

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

The cosine function is zero at odd multiples of π/2. Therefore:

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

Solving for x:

x = (2n + 1)π/12

This gives us an infinite set of critical points. For instance, when n = 0, x = π/12; when n = 1, x = 3π/12 = π/4; and so on. These points are where the function f(x) might change its direction. These critical points divide the x-axis into intervals. Within each interval, the function will either be increasing or decreasing. To determine this, we will perform a sign analysis of f'(x).

Conducting a Sign Analysis

Now we'll use a sign analysis to determine whether f'(x) is positive or negative in the intervals created by our critical points. We'll pick test points within each interval and evaluate f'(x) = 3cos(6x) at those points. This tells us the sign of the slope within that interval.

Consider the intervals: (-∞, π/12), (π/12, 3π/12), (3π/12, 5π/12), etc.

• Interval (-∞, π/12): Choose x = 0. f'(0) = 3cos(0) = 3 > 0. The function is increasing.
• Interval (π/12, 3π/12): Choose x = π/8. f'(π/8) = 3cos(6 * π/8) = 3cos(3π/4) < 0. The function is decreasing.
• Interval (3π/12, 5π/12): Choose x = π/3. f'(π/3) = 3cos(6 * π/3) = 3cos(2π) > 0. The function is increasing.

By continuing this process, we can determine the sign of f'(x) in each interval and identify where the function is increasing or decreasing. This sign analysis helps us understand the function's overall behavior. Keep in mind that the intervals will repeat due to the periodic nature of sine and cosine functions.

Determining Increasing and Decreasing Intervals

After performing the sign analysis, we can clearly define the intervals where f(x) = (1/2)sin(6x) is increasing and decreasing. Remember, f'(x) = 3cos(6x). Based on our sign analysis:

• Increasing Intervals: f(x) is increasing where f'(x) > 0. This occurs when x is in the intervals (π/12 + nπ/6, 5π/12 + nπ/6), where n is an integer.
• Decreasing Intervals: f(x) is decreasing where f'(x) < 0. This occurs when x is in the intervals (π/12 + nπ/6, 3π/12 + nπ/6), where n is an integer.

These intervals show us exactly where the function is going up and down. This information is crucial for sketching the graph, understanding the function's behavior over time, and applying these concepts to other real-world scenarios. We've used calculus to determine these intervals, providing a clear picture of the function's changing behavior. It's a fundamental concept that empowers us to analyze and predict the function's movements.

Summary and Conclusion

So, what have we learned, guys? We started with the function sin(3x)cos(3x), which we cleverly simplified to (1/2)sin(6x) using a double-angle identity. Then, we used calculus – specifically, the derivative – to find the critical points and analyze the function's behavior. We determined that the function is increasing in the intervals (π/12 + nπ/6, 5π/12 + nπ/6) and decreasing in the intervals (π/12 + nπ/6, 3π/12 + nπ/6). This understanding is key for anyone studying calculus or dealing with trigonometric functions in real-world applications. By mastering these concepts, you're not just solving a math problem, you're gaining the tools to understand dynamic systems, wave phenomena, and much more.

This kind of analysis is fundamental to understanding how functions behave. Knowing where a function increases or decreases allows us to sketch its graph accurately, identify local maxima and minima, and understand its overall trend. In conclusion, finding the increasing and decreasing intervals of sin(3x)cos(3x) involves simplifying the function, finding its derivative, identifying critical points, and conducting a sign analysis. This process offers insights into the function's behavior and is a valuable skill in calculus and its applications. Keep practicing, and you'll become a pro at analyzing these functions in no time! Keep up the great work and the exploring, guys! You've got this!