Increasing/Decreasing Intervals Of F(x) = √(x² + 1)

Hey everyone! Today, we're diving into a classic calculus problem: figuring out where the function $f(x) = \sqrt{x^2 + 1}$ is increasing and where it's decreasing. This involves a little bit of calculus magic, but don't worry, we'll break it down step by step so it's super easy to follow. So, grab your thinking caps, and let's get started!

Understanding Increasing and Decreasing Functions

Before we jump into the specifics of our function, let's quickly recap what it means for a function to be increasing or decreasing. In simple terms:

• Increasing Function: A function is increasing on an interval if its values go up as you move from left to right along the x-axis. Think of it like climbing a hill – as you move forward, you're going higher.
• Decreasing Function: Conversely, a function is decreasing if its values go down as you move from left to right. Imagine walking down a hill – as you move forward, you're going lower.

Calculus gives us a powerful tool to determine this behavior: the first derivative. The sign of the first derivative tells us whether the function is increasing or decreasing. If the first derivative is positive, the function is increasing. If it's negative, the function is decreasing. And if it's zero, we might have a critical point (a potential maximum or minimum).

So, to find the intervals where $f(x) = \sqrt{x^2 + 1}$ is increasing or decreasing, our mission is clear: we need to find its first derivative, figure out where it's positive, negative, and zero, and then interpret what that means for the function's behavior. This is the core concept we will be applying to solve this problem effectively. Remember, the derivative is the key to unlocking the secrets of a function's increasing and decreasing nature. Understanding this principle is crucial for mastering calculus. Let's dive into the steps!

Step 1: Finding the First Derivative

The first thing we need to do is find the derivative of our function, $f(x) = \sqrt{x^2 + 1}$. This involves using the chain rule, which is a fundamental concept in calculus for differentiating composite functions. The chain rule basically says that if you have a function inside another function, you differentiate the outer function, keeping the inner function the same, and then multiply by the derivative of the inner function.

In our case, we can think of $f(x)$ as having an outer function of the square root (which can be written as a power of 1/2) and an inner function of $x^2 + 1$. So, let's apply the chain rule:

1. Rewrite the square root: $f(x) = (x^2 + 1)^{1/2}$

2. Apply the power rule and chain rule: $f'(x) = \frac{1}{2}(x^2 + 1)^{-1/2} \cdot (2x)$

   Here, we brought down the exponent (1/2), decreased it by 1 (getting -1/2), and then multiplied by the derivative of the inner function $(x^2 + 1)$, which is $2x$.

3. Simplify: $f'(x) = \frac{2x}{2\sqrt{x^2 + 1}} = \frac{x}{\sqrt{x^2 + 1}}$

So, there we have it! The first derivative of our function is $f'(x) = \frac{x}{\sqrt{x^2 + 1}}$. This derivative is the key to understanding where our function is increasing or decreasing. This step highlights the importance of mastering differentiation techniques, especially the chain rule, in calculus. Now that we have the derivative, the next step is to analyze its sign to determine the intervals of increasing and decreasing behavior.

Step 2: Finding Critical Points

Critical points are the values of x where the derivative is either equal to zero or undefined. These points are crucial because they often mark the transition points where a function changes from increasing to decreasing, or vice versa. Critical points are like the potential turning points on a rollercoaster – they're where the ride might switch from going uphill to downhill.

Let's find the critical points for our derivative, $f'(x) = \frac{x}{\sqrt{x^2 + 1}}$:

1. Set the derivative equal to zero: $\frac{x}{\sqrt{x^2 + 1}} = 0$

   A fraction is equal to zero only if its numerator is zero. So, we have: $x = 0$

2. Find where the derivative is undefined:

   A fraction is undefined if its denominator is zero. So, we need to see if there are any values of x that make $\sqrt{x^2 + 1} = 0$.

   However, $x^2$ is always non-negative, so $x^2 + 1$ is always greater than or equal to 1. This means $\sqrt{x^2 + 1}$ is always greater than or equal to 1 and can never be zero.

Therefore, the only critical point we have is $x = 0$. This single critical point will help us divide the number line into intervals where we can analyze the sign of the derivative. Identifying critical points is a fundamental step in analyzing the behavior of a function. By finding where the derivative is zero or undefined, we pinpoint the potential locations of local maxima, local minima, or points where the function changes direction.

Step 3: Analyzing the Sign of the Derivative

Now that we've found our critical point ($x = 0$), we need to analyze the sign of the derivative $f'(x) = \frac{x}{\sqrt{x^2 + 1}}$ on the intervals created by this point. This is where we'll determine whether the function is increasing or decreasing in each interval. We'll create a sign chart, which is a visual tool to help us keep track of the sign of the derivative.

Our critical point divides the number line into two intervals: $(-\infty, 0)$ and $(0, \infty)$. We'll pick a test value within each interval and plug it into the derivative to see if the result is positive or negative.

1. Interval $(-\infty, 0)$: Let's pick a test value, say $x = -1$. $f'(-1) = \frac{-1}{\sqrt{(-1)^2 + 1}} = \frac{-1}{\sqrt{2}}$

   Since the result is negative, the derivative is negative on this interval, meaning the function is decreasing.

2. Interval $(0, \infty)$: Let's pick a test value, say $x = 1$. $f'(1) = \frac{1}{\sqrt{(1)^2 + 1}} = \frac{1}{\sqrt{2}}$

   Since the result is positive, the derivative is positive on this interval, meaning the function is increasing.

Analyzing the sign of the derivative is the core of determining increasing and decreasing intervals. This method allows us to systematically assess how the function's slope changes across its domain. By choosing test values in each interval, we can reliably infer the function's behavior within that interval. This step showcases the practical application of calculus in understanding the dynamics of functions.

Step 4: Determining Increasing and Decreasing Intervals

Based on our analysis of the sign of the derivative, we can now confidently state the intervals where the function $f(x) = \sqrt{x^2 + 1}$ is increasing and decreasing. This is the final step where we translate our mathematical findings into a clear description of the function's behavior.

• Decreasing Interval: The function is decreasing on the interval $(-\infty, 0)$ because $f'(x)$ is negative in this interval.
• Increasing Interval: The function is increasing on the interval $(0, \infty)$ because $f'(x)$ is positive in this interval.

Therefore, we've successfully determined that the function decreases until it reaches $x=0$, and then it starts increasing. This conclusion provides a comprehensive understanding of the function's trend. The ability to identify these intervals is crucial in various applications, such as optimization problems and curve sketching. This step demonstrates the power of calculus in providing precise insights into the behavior of functions.

Conclusion

So, there you have it! We've successfully found the intervals where the function $f(x) = \sqrt{x^2 + 1}$ is increasing and decreasing. We did this by finding the first derivative, identifying critical points, analyzing the sign of the derivative, and then interpreting the results. This process is a fundamental technique in calculus, and it's super useful for understanding the behavior of functions. Mastering these steps will greatly enhance your calculus skills. This example illustrates how calculus can be used to analyze and understand the behavior of functions, providing valuable insights for various applications in mathematics, science, and engineering. Keep practicing, and you'll become a pro at this in no time! If you guys have any questions, feel free to ask. Keep exploring the fascinating world of calculus! Remember, practice makes perfect, so keep applying these techniques to different functions.