Analyzing Increasing And Decreasing Intervals Of A Cosine Function

Oct 27, 2025 by ADMIN 67 views

Hey math enthusiasts! Let's dive into the fascinating world of calculus and figure out where the function f(x) = 4cos(2x) increases and decreases on the interval [-3π/2, 3π/2]. This is a super common problem, and understanding it will give you a solid foundation for more complex calculus concepts. So, grab your pencils, and let's get started!

Understanding the Basics: Increasing and Decreasing Functions

First things first, what does it even mean for a function to be increasing or decreasing? Well, think of it like climbing a hill. If you're going uphill as you move from left to right, the function is increasing. If you're going downhill, it's decreasing. More formally:

A function f(x) is increasing on an interval if, for any two points x1 and x2 in the interval, where x1 < x2, we have f(x1) < f(x2).
A function f(x) is decreasing on an interval if, for any two points x1 and x2 in the interval, where x1 < x2, we have f(x1) > f(x2).

Now, how do we actually figure this out mathematically? That's where derivatives come into play. The derivative of a function, denoted as f'(x), tells us the slope of the tangent line at any given point. If f'(x) > 0, the function is increasing. If f'(x) < 0, the function is decreasing. If f'(x) = 0, we have a critical point, where the function might change direction (could be a local max, min, or a saddle point). Remember, the derivative is our key to unlock this mystery! Understanding this connection between the derivative and the increasing/decreasing behavior is absolutely crucial. Don't worry if it doesn't click immediately; with practice, it'll become second nature. We are going to go through it step by step, so stick around!

Step 1: Find the Derivative of f(x)

Alright, let's get our hands dirty and calculate the derivative of our function, f(x) = 4cos(2x). We'll use the chain rule here, because we have a function within a function. The chain rule states that the derivative of cos(u) is -sin(u) * du/dx. Here’s how it breaks down:

Identify the inner and outer functions: The outer function is 4cos(u), and the inner function is u = 2x.
Differentiate the outer function: The derivative of 4cos(u) with respect to u is -4sin(u).
Differentiate the inner function: The derivative of u = 2x with respect to x is 2.
Apply the chain rule: Multiply the derivatives of the outer and inner functions: f'(x) = -4sin(2x) * 2 = -8sin(2x).

So, the derivative of f(x) = 4cos(2x) is f'(x) = -8sin(2x). This is the heart of the problem! We'll use this to find the intervals where the function is increasing or decreasing. Don't be intimidated by the chain rule; it just takes a little practice. With each example, you will feel more comfortable. Remember to take a little break if you get stuck, and then come back to it with fresh eyes.

Step 2: Find the Critical Points

Critical points are the x-values where the derivative is either equal to zero or undefined. These are the potential turning points of our function, where it could switch from increasing to decreasing or vice versa. Since our derivative is f'(x) = -8sin(2x), it's defined everywhere, so we only need to look for where it equals zero.

To find the critical points, we set f'(x) = 0:

-8sin(2x) = 0 sin(2x) = 0

Now, we need to find the values of 2x that make the sine function equal to zero. Remember the unit circle? The sine function is zero at multiples of π. Therefore:

2x = nπ, where n is an integer.

Solving for x, we get:

x = nπ/2

Now, we need to find the critical points within our given interval, [-3π/2, 3π/2]. Let's plug in different integer values for n to find the corresponding x values:

n = -3: x = (-3π)/2
n = -2: x = -π
n = -1: x = -π/2
n = 0: x = 0
n = 1: x = π/2
n = 2: x = π
n = 3: x = (3π)/2

So, our critical points within the interval are -3π/2, -π, -π/2, 0, π/2, π, and 3π/2. These points divide our interval into smaller subintervals where the function's behavior (increasing or decreasing) will be consistent. The critical points are our guide to understanding the function's journey!

Step 3: Determine the Sign of f'(x) in Each Interval

Now comes the fun part! We're going to test the sign of the derivative, f'(x) = -8sin(2x), in each subinterval created by our critical points. This will tell us whether the function is increasing (positive derivative) or decreasing (negative derivative).

Here's how we'll do it:

Create a number line: Draw a number line and mark all the critical points we found in Step 2: -3π/2, -π, -π/2, 0, π/2, π, and 3π/2. This will visually represent our intervals.
Choose a test value: In each interval, select a test value (any number within that interval). Avoid the critical points themselves.
Evaluate f'(x) at the test value: Plug the test value into the derivative, f'(x) = -8sin(2x).
Determine the sign: Determine whether the result is positive or negative. This tells us whether the function is increasing or decreasing in that interval.

Let's go through the intervals:

Interval 1: (-3π/2, -π): Choose x = -7π/4. f'(-7π/4) = -8sin(-7π/2) = -8(-1) = 8. Positive, so f(x) is increasing.
Interval 2: (-π, -π/2): Choose x = -3π/4. f'(-3π/4) = -8sin(-3π/2) = -8(1) = -8. Negative, so f(x) is decreasing.
Interval 3: (-π/2, 0): Choose x = -π/4. f'(-π/4) = -8sin(-π/2) = -8(-1) = 8. Positive, so f(x) is increasing.
Interval 4: (0, π/2): Choose x = π/4. f'(π/4) = -8sin(π/2) = -8(1) = -8. Negative, so f(x) is decreasing.
Interval 5: (π/2, π): Choose x = 3π/4. f'(3π/4) = -8sin(3π/2) = -8(-1) = 8. Positive, so f(x) is increasing.
Interval 6: (π, 3π/2): Choose x = 5π/4. f'(5π/4) = -8sin(5π/2) = -8(1) = -8. Negative, so f(x) is decreasing.

Now, we have a clear picture of how the function behaves in each interval. This sign analysis is a cornerstone of understanding the function's ups and downs.

Step 4: State the Intervals of Increase and Decrease

Based on our analysis in Step 3, we can now state the intervals where f(x) = 4cos(2x) is increasing and decreasing:

Increasing Intervals: (-3π/2, -π), (-π/2, 0), (π/2, π)
Decreasing Intervals: (-π, -π/2), (0, π/2), (π, 3π/2)

And there you have it! We've successfully analyzed the increasing and decreasing intervals of our cosine function. You can visualize this by sketching the graph of the function; you'll see it climbing in the increasing intervals and falling in the decreasing intervals. Excellent work! With each problem you solve, you are building a stronger understanding of calculus concepts.

Conclusion

So, to recap, we:

Found the derivative of the function using the chain rule.
Identified the critical points by setting the derivative to zero and solving for x.
Analyzed the sign of the derivative in each interval using test values.
Stated the intervals where the function is increasing and decreasing.

Understanding increasing and decreasing intervals is fundamental in calculus. This skill is critical for applications like optimization problems where you need to find maximum or minimum values. Keep practicing, and you'll become a pro in no time! Keep up the great work, and happy calculating!