How To Find Where A Function Is Increasing And Decreasing

The concept of increasing and decreasing functions is fundamental in calculus and analysis, providing insights into the behavior and characteristics of mathematical functions. Understanding where a function is increasing or decreasing is crucial for sketching graphs, finding local extrema (maxima and minima), and solving optimization problems. This article delves into the methods for determining the intervals where a function is increasing or decreasing, complete with examples and explanations.

Understanding Increasing and Decreasing Functions

A function is said to be increasing on an interval if, for any two points x₁ and x₂ in that interval with x₁ < x₂, we have f(x₁) ≤ f(x₂). If f(x₁) < f(x₂), the function is strictly increasing.

Conversely, a function is decreasing on an interval if, for any two points x₁ and x₂ in that interval with x₁ < x₂, we have f(x₁) ≥ f(x₂). If f(x₁) > f(x₂), the function is strictly decreasing.

The Role of the Derivative

The derivative of a function, denoted as f'(x), plays a pivotal role in determining where a function is increasing or decreasing. The sign of the derivative provides this information:

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

Critical Points

Critical points are points in the domain of the function where the derivative is either zero or undefined. These points are essential because they mark potential locations where the function changes from increasing to decreasing or vice versa. Critical points are found by solving the equation f'(x) = 0 or identifying where f'(x) is undefined.

Steps to Find Where a Function Is Increasing and Decreasing

To determine the intervals where a function is increasing and decreasing, follow these steps:

1. Find the Derivative: Compute the derivative of the function f(x), denoted as f'(x).
2. Find Critical Points: Determine all critical points by finding the values of x for which f'(x) = 0 or f'(x) is undefined.
3. Create Intervals: Use the critical points to divide the domain of f(x) into intervals.
4. Test Intervals: Choose a test value c within each interval and evaluate f'(c).
   • If f'(c) > 0, the function is increasing on that interval.
   • If f'(c) < 0, the function is decreasing on that interval.
   • If f'(c) = 0, the function is constant on that interval.
5. Conclusion: State the intervals where the function is increasing, decreasing, or constant.

Step-by-Step Explanation with Examples

Let's illustrate this process with several examples.

Example 1: Finding Increasing and Decreasing Intervals

Consider the function f(x) = x³ - 3x² - 9x + 5.

1. Find the Derivative:
   • f'(x) = 3x² - 6x - 9
2. Find Critical Points:
   • Set f'(x) = 0:
     • 3x² - 6x - 9 = 0
     • Divide by 3: x² - 2x - 3 = 0
     • Factor: (x - 3)(x + 1) = 0
     • Solve for x: x = 3, x = -1
   • The critical points are x = -1 and x = 3.
3. Create Intervals:
   • The critical points divide the real number line into three intervals:
     • (−∞, −1)
     • (-1, 3)
     • (3, ∞)
4. Test Intervals:
   • Choose a test value in each interval:
     • For (−∞, −1), let c = -2:
       • f'(-2) = 3(-2)² - 6(-2) - 9 = 12 + 12 - 9 = 15 > 0. Thus, f(x) is increasing on (−∞, −1).
     • For (-1, 3), let c = 0:
       • f'(0) = 3(0)² - 6(0) - 9 = -9 < 0. Thus, f(x) is decreasing on (-1, 3).
     • For (3, ∞), let c = 4:
       • f'(4) = 3(4)² - 6(4) - 9 = 48 - 24 - 9 = 15 > 0. Thus, f(x) is increasing on (3, ∞).
5. Conclusion:
   • f(x) is increasing on (−∞, −1) and (3, ∞).
   • f(x) is decreasing on (-1, 3).

Example 2: Dealing with Undefined Derivatives

Consider the function f(x) = x^(2/3).

1. Find the Derivative:
   • f'(x) = (2/3)x^(-1/3) = 2 / (3 * x^(1/3))
2. Find Critical Points:
   • Set f'(x) = 0:
     • 2 / (3 * x^(1/3)) = 0 has no solution.
   • Find where f'(x) is undefined:
     • f'(x) is undefined when x = 0.
   • The critical point is x = 0.
3. Create Intervals:
   • The critical point divides the real number line into two intervals:
     • (−∞, 0)
     • (0, ∞)
4. Test Intervals:
   • Choose a test value in each interval:
     • For (−∞, 0), let c = -1:
       • f'(-1) = 2 / (3 * (-1)^(1/3)) = 2 / -3 < 0. Thus, f(x) is decreasing on (−∞, 0).
     • For (0, ∞), let c = 1:
       • f'(1) = 2 / (3 * (1)^(1/3)) = 2 / 3 > 0. Thus, f(x) is increasing on (0, ∞).
5. Conclusion:
   • f(x) is decreasing on (−∞, 0).
   • f(x) is increasing on (0, ∞).

Example 3: Trigonometric Function

Consider the function f(x) = sin(x) on the interval [0, 2π].

1. Find the Derivative:
   • f'(x) = cos(x)
2. Find Critical Points:
   • Set f'(x) = 0:
     • cos(x) = 0
     • The solutions in the interval [0, 2π] are x = π/2 and x = 3π/2.
3. Create Intervals:
   • The critical points divide the interval [0, 2π] into three intervals:
     • [0, π/2)
     • (π/2, 3π/2)
     • (3π/2, 2π]
4. Test Intervals:
   • Choose a test value in each interval:
     • For [0, π/2), let c = π/4:
       • f'(π/4) = cos(π/4) = √2/2 > 0. Thus, f(x) is increasing on [0, π/2).
     • For (π/2, 3π/2), let c = π:
       • f'(π) = cos(π) = -1 < 0. Thus, f(x) is decreasing on (π/2, 3π/2).
     • For (3π/2, 2π], let c = 7π/4:
       • f'(7π/4) = cos(7π/4) = √2/2 > 0. Thus, f(x) is increasing on (3π/2, 2π].
5. Conclusion:
   • f(x) is increasing on [0, π/2) and (3π/2, 2π].
   • f(x) is decreasing on (π/2, 3π/2).

Advanced Techniques and Considerations

Second Derivative Test

The second derivative test is used to determine whether a critical point is a local maximum or a local minimum. If f'(c) = 0 and f''(c) > 0, then f(x) has a local minimum at x = c. If f'(c) = 0 and f''(c) < 0, then f(x) has a local maximum at x = c. If f''(c) = 0, the test is inconclusive.

Functions with Discontinuities

When dealing with functions that have discontinuities, it's important to consider the intervals created by these discontinuities in addition to the critical points. The function may change from increasing to decreasing (or vice versa) at a point of discontinuity.

Absolute Maxima and Minima

To find the absolute maxima and minima of a function on a closed interval, evaluate the function at the critical points and at the endpoints of the interval. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Applications

The concepts of increasing and decreasing functions have numerous applications in various fields:

• Optimization Problems: Finding the maximum or minimum value of a function is crucial in optimization problems, such as maximizing profit or minimizing cost.
• Graphing Functions: Understanding where a function is increasing or decreasing helps in sketching accurate graphs.
• Economics: Analyzing cost and revenue functions to determine optimal production levels.
• Physics: Studying motion and velocity functions to understand acceleration and deceleration.
• Engineering: Designing structures and systems that optimize performance and efficiency.

Common Mistakes to Avoid

• Forgetting Critical Points: Make sure to find all critical points, including those where the derivative is zero and those where it is undefined.
• Incorrectly Calculating the Derivative: Double-check the derivative calculation to avoid errors.
• Ignoring Endpoints: When finding absolute maxima and minima on a closed interval, remember to check the endpoints.
• Misinterpreting the Sign of the Derivative: Be careful when determining whether the function is increasing or decreasing based on the sign of the derivative.

Conclusion

Determining where a function is increasing or decreasing is a fundamental skill in calculus and analysis. By following the steps outlined in this article, you can confidently analyze the behavior of functions and gain valuable insights into their properties. Remember to find the derivative, identify critical points, create intervals, test values, and draw appropriate conclusions. With practice and attention to detail, you can master this essential technique and apply it to a wide range of problems. Understanding these concepts provides a solid foundation for further study in mathematics and its applications in various fields.