The Function Graphed Above Is Decreasing On The Interval

The Function Graphed Above is Decreasing on the Interval: A Comprehensive Guide

Understanding the intervals where a function is increasing or decreasing is fundamental in calculus and crucial for analyzing the behavior of functions. This comprehensive guide delves deep into identifying decreasing intervals, providing a step-by-step approach, practical examples, and tackling various function types. We'll explore how to determine these intervals using various techniques, including analyzing graphs, derivatives, and applying the first derivative test.

Understanding Increasing and Decreasing Functions

A function is considered increasing on an interval if, for any two points x₁ and x₂ within that interval where x₁ < x₂, f(x₁) < f(x₂). Visually, this means the graph of the function is rising as you move from left to right. Conversely, a function is decreasing on an interval if, for any two points x₁ and x₂ within that interval where x₁ < x₂, f(x₁) > f(x₂). Graphically, this represents a falling graph from left to right. A function can be increasing, decreasing, or neither (constant) over different intervals of its domain.

Identifying Decreasing Intervals from a Graph

The simplest way to identify decreasing intervals is by visually inspecting the graph of the function. Look for sections of the graph where the function's value consistently decreases as the x-value increases.

Example: Imagine a graph showing a curve that slopes downward from x = -2 to x = 1. This visually indicates that the function is decreasing on the interval (-2, 1). Note that parentheses are used to indicate that the endpoints are not included in the decreasing interval. We'll delve into the reasons behind this later when discussing open and closed intervals.

Key Considerations When Using Graphs:

• Scale: Ensure you accurately understand the scale of the axes to avoid misinterpreting the slope.
• Local vs. Global: A function might decrease on a small interval (local) while increasing overall (globally). Pay attention to the specific interval under consideration.
• Asymptotes: Be mindful of vertical asymptotes, where the function may approach infinity or negative infinity. These points usually aren't included in decreasing intervals.

Using the First Derivative Test

While visual inspection is helpful, it's not always precise, especially with complex functions. The first derivative test provides a robust and analytical method for determining increasing and decreasing intervals.

The first derivative of a function, f'(x), represents the instantaneous rate of change of the function at any point x.

• If f'(x) > 0 on an interval, then f(x) is increasing on that interval.
• If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.
• If f'(x) = 0 on an interval, then f(x) is constant on that interval.

Steps to apply the first derivative test:

1. Find the first derivative, f'(x), of the function f(x). This often involves using rules of differentiation like the power rule, product rule, quotient rule, or chain rule.

2. Find the critical points. Critical points are values of x where f'(x) = 0 or f'(x) is undefined. These points potentially mark the boundaries between increasing and decreasing intervals.

3. Analyze the sign of f'(x) in the intervals determined by the critical points. Test points within each interval to determine whether f'(x) is positive or negative. A positive sign indicates an increasing interval, and a negative sign indicates a decreasing interval.

Example: Let's consider the function f(x) = x³ - 3x.

1. First Derivative: f'(x) = 3x² - 3

2. Critical Points: Set f'(x) = 0: 3x² - 3 = 0 => x² = 1 => x = ±1. These are our critical points.

3. Interval Analysis:

   • Interval (-∞, -1): Choose a test point, say x = -2. f'(-2) = 3(-2)² - 3 = 9 > 0. Therefore, f(x) is increasing on (-∞, -1).

   • Interval (-1, 1): Choose a test point, say x = 0. f'(0) = 3(0)² - 3 = -3 < 0. Therefore, f(x) is decreasing on (-1, 1).

   • Interval (1, ∞): Choose a test point, say x = 2. f'(2) = 3(2)² - 3 = 9 > 0. Therefore, f(x) is increasing on (1, ∞).

Therefore, the function f(x) = x³ - 3x is decreasing on the interval (-1, 1).

Dealing with Different Function Types

The techniques discussed above are applicable to various function types, including polynomial, rational, exponential, logarithmic, and trigonometric functions. However, the complexity of finding the derivative and analyzing the intervals may vary.

Polynomial Functions

Polynomial functions are relatively straightforward. Their derivatives are also polynomials, making finding critical points and analyzing intervals simpler.

Rational Functions

Rational functions (ratios of polynomials) require the quotient rule for differentiation. Remember to consider points where the denominator is zero, as these are potential points of discontinuity and may affect the increasing/decreasing intervals.

Exponential and Logarithmic Functions

The derivatives of exponential and logarithmic functions involve the natural logarithm (ln) and the exponential function (eˣ). Pay close attention to the properties of these functions when analyzing the sign of the derivative.

Trigonometric Functions

Trigonometric functions require knowledge of trigonometric derivatives. Their periodic nature means increasing/decreasing intervals repeat.

Open vs. Closed Intervals

The notation for intervals is crucial. Parentheses () denote open intervals, meaning the endpoints are not included. Brackets [] denote closed intervals, meaning the endpoints are included. When determining increasing/decreasing intervals, we usually use open intervals because the function's behavior at the critical points is often undefined or doesn't directly reflect increasing/decreasing behavior. The function might be neither increasing nor decreasing precisely at a critical point; it changes its behavior around that point.

Second Derivative Test and Concavity

While the first derivative test identifies increasing and decreasing intervals, the second derivative test provides additional information about the function's concavity. The second derivative, f''(x), determines whether the function is concave up (f''(x) > 0) or concave down (f''(x) < 0). This information can help refine your understanding of the function's behavior.

Applications of Increasing/Decreasing Intervals

Understanding increasing and decreasing intervals has several practical applications:

• Optimization Problems: Finding maximum and minimum values of a function is often crucial in optimization problems (e.g., maximizing profit, minimizing cost). These extrema occur at points where the function transitions from increasing to decreasing or vice versa.

• Modeling Real-World Phenomena: Many real-world phenomena can be modeled using functions, and analyzing their increasing/decreasing intervals provides valuable insights into the behavior of the phenomenon over time or other variables.

• Sketching Graphs: Identifying increasing and decreasing intervals helps in accurately sketching the graph of a function.

• Analyzing Rates of Change: The derivative directly shows the rate of change of a function. The sign of the derivative informs about whether that rate is positive (increasing) or negative (decreasing).

Conclusion

Determining the intervals where a function is decreasing is a fundamental concept in calculus with broad applications. Whether using graphical analysis or the more rigorous first derivative test, understanding how to identify these intervals provides key insights into a function's behavior and is essential for tackling a wide range of mathematical and real-world problems. Remember to practice regularly with diverse functions to build your proficiency and understanding. The more you practice, the more intuitive this process will become, making you a more confident and skilled problem-solver.