Determining If F(x) = 4x³ + 2x² - 3 Is Increasing On [-½, 1]

Hey guys! Today, we're diving deep into a calculus question that might pop up on your exams. We're going to analyze whether the function f(x) = 4x³ + 2x² - 3 is increasing on the interval [-½, 1]. This involves some cool calculus concepts, and I'll walk you through each step to make sure you get it. So, grab your pencils, and let's get started!

Understanding Increasing Functions

Before we jump into the specific function, let's nail down what it means for a function to be increasing. In simple terms, a function is increasing on an interval if its y-values go up as its x-values increase. Think of it like climbing a hill – as you move to the right (increase x), you're going uphill (increase y).

But how do we show this mathematically? That's where calculus comes in handy! The derivative of a function, f'(x), tells us the slope of the function at any point. If the derivative is positive (f'(x) > 0), it means the function is sloping upwards – hence, it's increasing. If the derivative is negative (f'(x) < 0), the function is sloping downwards and decreasing. And if the derivative is zero (f'(x) = 0), we have a horizontal tangent, which could be a maximum, a minimum, or a point of inflection.

So, to figure out if our function f(x) = 4x³ + 2x² - 3 is increasing on [-½, 1], we need to find its derivative and see if it's positive on that interval. This involves understanding the fundamental concept of derivatives and their relationship to the function's behavior. We're not just plugging in numbers; we're understanding the underlying principles of calculus.

Moreover, we need to consider the endpoints of the interval and any critical points within the interval. Critical points are where the derivative is either zero or undefined. These points are crucial because they can mark changes in the function's behavior – from increasing to decreasing, or vice versa. We'll analyze these points carefully to ensure a comprehensive understanding of the function's behavior on the given interval.

Finding the Derivative of f(x)

Okay, let's get our hands dirty with some actual calculus. Our function is f(x) = 4x³ + 2x² - 3. To find its derivative, f'(x), we'll use the power rule. Remember, the power rule states that if f(x) = axⁿ, then f'(x) = naxⁿ⁻¹.

Let's break it down step-by-step:

• For the term 4x³, we have a = 4 and n = 3. Applying the power rule, we get 3 * 4x³⁻¹ = 12x².
• For the term 2x², we have a = 2 and n = 2. Applying the power rule, we get 2 * 2x²⁻¹ = 4x.
• For the constant term -3, the derivative is 0 because the derivative of a constant is always zero.

So, putting it all together, the derivative of f(x) = 4x³ + 2x² - 3 is f'(x) = 12x² + 4x. Awesome! We've got the derivative. Now, we need to figure out where this derivative is positive on the interval [-½, 1]. This is a crucial step because it tells us where the original function is increasing. Finding the derivative is just the first part; the real analysis comes in understanding what this derivative tells us about the function's behavior.

To fully understand the derivative, we might also want to consider its graphical representation. The graph of f'(x) = 12x² + 4x is a parabola, and by analyzing this parabola, we can visually see where the derivative is positive, negative, or zero. This graphical perspective can provide additional insights into the behavior of the original function.

Analyzing the Sign of f'(x) on [-½, 1]

Now comes the fun part – figuring out where our derivative, f'(x) = 12x² + 4x, is positive on the interval [-½, 1]. Remember, a positive derivative means the function is increasing. So, we're on the hunt for where 12x² + 4x > 0.

First, let's find the critical points. These are the points where f'(x) = 0 or is undefined. In this case, f'(x) is a polynomial, so it's defined everywhere. We just need to find where it's zero:

12x² + 4x = 0

We can factor out a 4x:

4x(3x + 1) = 0

This gives us two critical points: x = 0 and x = -⅓.

Now, we'll use these critical points to divide our interval [-½, 1] into subintervals and test the sign of f'(x) in each subinterval. Our subintervals are [-½, -⅓), (-⅓, 0), and (0, 1]. This method helps us determine the intervals where the function is increasing or decreasing.

• On [-½, -⅓): Let's pick a test point, say x = -0.4. Plugging into f'(x), we get 12(-0.4)² + 4(-0.4) = 1.92 - 1.6 = 0.32, which is positive. So, f(x) is increasing on this interval.
• On (-⅓, 0): Let's pick a test point, say x = -0.1. Plugging into f'(x), we get 12(-0.1)² + 4(-0.1) = 0.12 - 0.4 = -0.28, which is negative. So, f(x) is decreasing on this interval.
• On (0, 1]: Let's pick a test point, say x = 0.5. Plugging into f'(x), we get 12(0.5)² + 4(0.5) = 3 + 2 = 5, which is positive. So, f(x) is increasing on this interval.

By analyzing these intervals, we get a clear picture of how the function behaves. This step is vital for understanding the function's overall trend and answering our main question.

Conclusion: Is f(x) Increasing on [-½, 1]?

Alright, we've done the heavy lifting! We found the derivative, f'(x) = 12x² + 4x, and we analyzed its sign on the interval [-½, 1]. We found that f'(x) is positive on [-½, -⅓), negative on (-⅓, 0), and positive on (0, 1].

So, the answer to our question is: No, f(x) is not increasing on the entire interval [-½, 1]. It's increasing on [-½, -⅓) and (0, 1], but it's decreasing on (-⅓, 0). This means that the function has a mix of increasing and decreasing behavior within the given interval.

This comprehensive analysis shows how we can use calculus to understand the behavior of a function. By finding the derivative and analyzing its sign, we can determine where a function is increasing, decreasing, or staying constant. This is a fundamental skill in calculus and is essential for solving many types of problems.

Key Takeaways:

• To determine if a function is increasing on an interval, find its derivative.
• If the derivative is positive, the function is increasing.
• If the derivative is negative, the function is decreasing.
• Find critical points (where the derivative is zero or undefined) to divide the interval into subintervals.
• Test the sign of the derivative in each subinterval to determine the function's behavior.

I hope this detailed explanation helps you guys understand how to analyze the increasing and decreasing behavior of functions. Keep practicing, and you'll become calculus pros in no time! If you have any questions, feel free to ask. Keep rocking!