Finding Intervals Where A Function Is Decreasing

Hey everyone! Today, we're diving into a cool math problem where we figure out where a function is decreasing. We'll use calculus to solve this, so if you're into that, awesome! If not, don't sweat it – I'll explain everything clearly. So, let's get started. The core idea here is to understand the relationship between a function's derivative and its increasing or decreasing behavior. This is like, super fundamental in calculus, and once you get it, you can solve a bunch of problems.

Understanding the Problem: The Basics

Okay, so the problem gives us a function, let's call it $h$, but it's not defined at zero. That's fine; it just means we have to be careful around $x = 0$. We're also given its derivative, $h'(x) = \frac{x+3}{x^2}$. Remember, the derivative tells us the slope of the function at any point. Our mission, should we choose to accept it, is to find the intervals where $h$ is decreasing. A function is decreasing when its slope is negative. Thus, we need to figure out where $h'(x) < 0$.

Let's break that down, because it's the most important part of the problem. If the derivative $h'(x)$ is negative, it means the function $h$ is going downhill at that point. If $h'(x)$ is positive, the function is going uphill. When $h'(x)$ is zero, it could be a turning point, like a peak or a valley. So, our strategy is to analyze the derivative's sign. Think of it like this: the derivative is the map, and we want to know which roads are sloping downwards. To do that, we have to look closely at the equation $h'(x) = \frac{x+3}{x^2}$. Because the denominator is $x^2$, it's always positive (unless x=0, where it is undefined). Therefore, the sign of the derivative depends on the numerator, $x+3$. If $x+3$ is negative, then the whole fraction is negative and the function is decreasing. If $x+3$ is positive, then the fraction is positive and the function is increasing. Got it? Awesome.

Now, let's see how we can tackle this problem step by step, so even if you're new to calculus, you can easily follow along. I'll make sure to explain everything thoroughly, so you don't feel lost at any point. We'll start by analyzing the derivative to determine the intervals of increase and decrease. Then, we will create a sign chart, which is a visual tool that helps us organize our analysis. Finally, we will provide the final answer, which are the intervals where the function is decreasing. Let's start with the first step which consists in analyzing the derivative. Ready?

Step 1: Analyzing the Derivative $h'(x) = \frac{x+3}{x^2}$

Alright, let's analyze the derivative $h'(x) = \frac{x+3}{x^2}$. We already know that $x^2$ is always positive (except at $x=0$, where the function is undefined). Therefore, the sign of $h'(x)$ depends solely on the numerator, $x+3$. So, we want to know when $x+3 < 0$, because this is when the function is decreasing. This occurs when $x < -3$. The critical points are where the derivative is either zero or undefined. In our case, the derivative is zero when $x+3 = 0$, which means $x = -3$. It's undefined when $x = 0$. These points are important because they divide the number line into intervals where the function's behavior might change. Think of them as the landmarks we need to navigate the landscape of the function.

Now, let's look at the intervals: $(-\infty, -3)$, $(-3, 0)$, and $(0, \infty)$. In the interval $(-\infty, -3)$, $x+3$ is negative and $x^2$ is positive, so $h'(x)$ is negative, which means the function is decreasing. In the interval $(-3, 0)$, $x+3$ is positive, and $x^2$ is positive, making $h'(x)$ positive, which means the function is increasing. Finally, in the interval $(0, \infty)$, $x+3$ is positive and $x^2$ is positive, making $h'(x)$ positive, meaning the function is increasing. This kind of analysis is super common when you're trying to understand how functions behave. Knowing these intervals helps us sketch the graph or understand its behavior without having to plot a bunch of points. It's really the heart of how you use calculus to understand the shapes and behaviors of functions.

Now, that we know how to analyze a derivative let's create a sign chart that will help us keep track of all this information in an organized way.

Step 2: Creating a Sign Chart

Creating a sign chart is like setting up a cheat sheet for our derivative analysis. It's a visual tool that organizes the information we found in Step 1. The sign chart helps us keep track of where the function is increasing or decreasing. Basically, we put the critical points ($x=-3$ and $x=0$) on a number line and test the sign of the derivative in each interval. This gives us a quick way to see where the function is increasing or decreasing.

Let's construct the sign chart: Draw a number line. Mark the critical points, $x=-3$ and $x=0$. These points divide the number line into intervals: $(-\infty, -3)$, $(-3, 0)$, and $(0, \infty)$. Now, we will test the sign of $h'(x)$ in each interval. For the interval $(-\infty, -3)$, we pick a test point, say $x=-4$. Then, $h'(-4) = \frac{-4+3}{(-4)^2} = \frac{-1}{16}$, which is negative. This means $h(x)$ is decreasing in this interval. For the interval $(-3, 0)$, pick $x=-1$. Then, $h'(-1) = \frac{-1+3}{(-1)^2} = \frac{2}{1} = 2$, which is positive. So, $h(x)$ is increasing. For the interval $(0, \infty)$, pick $x=1$. Then, $h'(1) = \frac{1+3}{1^2} = 4$, which is positive, meaning $h(x)$ is increasing. We note these signs on the number line to visually represent where $h(x)$ is increasing or decreasing. The sign chart will clearly indicate the intervals where $h(x)$ is decreasing. This makes it super easy to spot the decreasing intervals without having to re-calculate everything constantly. It's a handy tool for understanding function behavior.

So, to recap the sign chart: In the interval $(-\infty, -3)$, $h'(x) < 0$, the function is decreasing. In the interval $(-3, 0)$, $h'(x) > 0$, the function is increasing. In the interval $(0, \infty)$, $h'(x) > 0$, the function is increasing. Next, let's provide our final answer.

Step 3: Determining the Intervals Where $h$ is Decreasing

Alright, guys, based on our analysis and sign chart, we're ready to nail down the intervals where the function $h$ is decreasing. Remember, the function is decreasing when $h'(x) < 0$. Looking back at our sign chart, we found that $h'(x)$ is negative in the interval $(-\infty, -3)$. Thus, the function $h$ is decreasing in the interval $(-\infty, -3)$. We exclude $x=0$ because the original function $h(x)$ is not defined at this point. Also, remember that a function cannot decrease at a single point, so we just focus on the intervals. Therefore, the answer is: The function $h$ is decreasing on the interval $(-\infty, -3)$.

That's it, folks! We've successfully found the interval where the function $h$ is decreasing. We did it by analyzing the derivative, creating a sign chart, and interpreting the results. This problem demonstrates a fundamental concept in calculus: the relationship between a function's derivative and its behavior. Understanding this link is key to solving many calculus problems.

To recap: 1. We found the derivative. 2. We analyzed the sign of the derivative. 3. We created a sign chart. 4. We determined the intervals where $h'(x) < 0$. This is a powerful technique for understanding the behavior of functions. I hope this explanation was helpful. If you have any questions, feel free to ask. Keep practicing, and you'll get the hang of it! Good job, everyone! And remember, practice makes perfect. Keep doing problems like this, and you'll become a pro in no time.