Decreasing Intervals Of A Cubic Function: A Step-by-Step Guide

by Andrew McMorgan 63 views

Hey Plastik Magazine readers! Let's dive into some calculus today, shall we? We're going to tackle a classic problem: finding the intervals where a function is decreasing. Specifically, we'll work with the cubic function f(x) = -2x³ + 3x² + 12x + 35. Don't worry if this sounds intimidating; we'll break it down into easy-to-follow steps. This is a fundamental concept in calculus, essential for understanding the behavior of functions. By the end of this guide, you'll be able to confidently determine where a function is increasing or decreasing.

Understanding the Problem and Key Concepts

Finding the decreasing intervals of a function is all about figuring out where the function's graph slopes downwards as you move from left to right. This is where the derivative comes into play. The derivative of a function, denoted as f'(x), tells us the slope of the function at any given point. If f'(x) > 0, the function is increasing. If f'(x) < 0, the function is decreasing. And if f'(x) = 0, we have a critical point, which could be a local maximum, a local minimum, or a saddle point. The concept is pretty straightforward: when the slope is negative, the function is going down; that's our decreasing interval!

Critical points are crucial because they mark the potential boundaries between increasing and decreasing intervals. These are the points where the function changes direction. They are found by setting the derivative equal to zero and solving for x. Furthermore, the first derivative test is a method used to determine whether a critical point is a local maximum, a local minimum, or neither. The test involves analyzing the sign of the derivative to the left and right of the critical points.

Now, let's get our hands dirty with our specific function, f(x) = -2x³ + 3x² + 12x + 35. We will go through the steps of finding its derivative, identifying critical points, and determining the intervals where the derivative is negative. This approach applies to various functions, making this skill highly valuable in mathematics and beyond. This is more than just solving a math problem; it's about gaining a deeper understanding of how functions behave and how to analyze their properties using calculus. So, let’s get started, shall we?

Step-by-Step Solution

Alright, buckle up, guys! We're going to find where f(x) = -2x³ + 3x² + 12x + 35 is decreasing. Here’s a detailed, step-by-step breakdown to guide us through:

Step 1: Find the Derivative, f'(x)

The first step is always to find the derivative of the function. This gives us the slope of the tangent line at any point x. Using the power rule, which states that the derivative of xⁿ is nxⁿ⁻¹, we differentiate each term in f(x).

  • The derivative of -2x³ is -6x².
  • The derivative of 3x² is 6x.
  • The derivative of 12x is 12.
  • The derivative of the constant 35 is 0.

So, our derivative is f'(x) = -6x² + 6x + 12. Awesome, we're already halfway there!

Step 2: Find the Critical Points

Next up, we need to find the critical points. These are the x-values where f'(x) = 0 or where the derivative is undefined (although it's not applicable in this case because this is a polynomial function). So, let's set f'(x) equal to zero and solve for x:

-6x² + 6x + 12 = 0

To simplify, let's divide the entire equation by -6:

x² - x - 2 = 0

Now, factor the quadratic equation:

(x - 2)(x + 1) = 0

This gives us two critical points: x = 2 and x = -1. These are the x-values where the function f(x) might change direction – cool, right?

Step 3: Determine the Intervals of Increasing/Decreasing

Now we'll use these critical points to divide the number line into intervals. Our critical points are -1 and 2, which create three intervals:

  • Interval 1: (-∞, -1)
  • Interval 2: (-1, 2)
  • Interval 3: (2, ∞)

To determine whether f(x) is increasing or decreasing in each interval, we'll test a value within each interval in our derivative, f'(x) = -6x² + 6x + 12. If f'(x) > 0, the function is increasing; if f'(x) < 0, it's decreasing. Let's do this!

  • Interval (-∞, -1): Let's test x = -2. f'(-2) = -6(-2)² + 6(-2) + 12 = -24 - 12 + 12 = -24. Since f'(-2) < 0, the function is decreasing in this interval.
  • Interval (-1, 2): Let's test x = 0. f'(0) = -6(0)² + 6(0) + 12 = 12. Since f'(0) > 0, the function is increasing in this interval.
  • Interval (2, ∞): Let's test x = 3. f'(3) = -6(3)² + 6(3) + 12 = -54 + 18 + 12 = -24. Since f'(3) < 0, the function is decreasing in this interval.

Step 4: State the Decreasing Intervals

Based on our calculations, f(x) is decreasing in the intervals where f'(x) < 0. Looking back at our tests, we found that f'(x) is negative in the intervals (-∞, -1) and (2, ∞). Therefore, the function f(x) = -2x³ + 3x² + 12x + 35 is decreasing on the intervals (-∞, -1) and (2, ∞).

Conclusion: Summarizing Our Findings

Alright, folks, we've successfully navigated the process of finding the decreasing intervals of the cubic function f(x) = -2x³ + 3x² + 12x + 35. We found that the function decreases on the intervals (-∞, -1) and (2, ∞). This means that as x moves from negative infinity up to -1, and then from 2 to positive infinity, the value of the function f(x) goes down. The function increases between x = -1 and x = 2. Remember, the key steps were:

  1. Finding the derivative, f'(x).
  2. Determining critical points (where f'(x) = 0).
  3. Testing intervals to determine the sign of f'(x).
  4. Identifying the intervals where f'(x) < 0.

Understanding how to determine increasing and decreasing intervals is a cornerstone of calculus. It helps us understand the behavior of functions and visualize their graphs. This knowledge is not just useful for math exams; it has applications in fields like physics, economics, and engineering, where understanding change and rates of change is crucial. Keep practicing these concepts, and you’ll master them in no time! So, keep exploring the fascinating world of calculus, guys. You've got this!