On What Interval Is F Increasing

Let's explore how to pinpoint the intervals where a function f is increasing. This involves understanding the relationship between a function's derivative and its increasing or decreasing behavior, and applying calculus techniques to find those intervals precisely.

Understanding Increasing Functions

A function f is considered increasing on an interval if, for any two points x₁ and x₂ in that interval, where x₁ < x₂, we have f(x₁) < f(x₂). Simply put, as the input x increases, the output f(x) also increases. Visualizing the graph of an increasing function, we see it climbs upwards from left to right within that specific interval.

The Derivative's Role: A Key Indicator

The derivative of a function, denoted as f'(x), provides invaluable information about the function's rate of change. Specifically:

• If f'(x) > 0 on an interval, the function f is increasing on that interval.
• If f'(x) < 0 on an interval, the function f is decreasing on that interval.
• If f'(x) = 0 on an interval, the function f is constant on that interval. Points where f'(x) = 0 are known as critical points, and they often mark the transition points between increasing and decreasing intervals (or vice-versa).

Steps to Determine Intervals Where f is Increasing

Here's a systematic approach to determine the intervals where a function f is increasing:

1. Find the Derivative: Calculate the derivative of the function, f'(x). This is the foundational step. Utilize the rules of differentiation (power rule, product rule, quotient rule, chain rule, etc.) appropriately.

2. Find Critical Points: Determine the critical points of the function. Critical points are the values of x where f'(x) = 0 or where f'(x) is undefined. These points are crucial because they often (though not always) separate intervals of increasing and decreasing behavior.

   • Solving f'(x) = 0: Set the derivative equal to zero and solve for x. These are the points where the tangent line to the curve is horizontal.
   • Points where f'(x) is undefined: Check for any values of x where the derivative does not exist. This often occurs when the derivative has a denominator that equals zero (leading to division by zero), or in piecewise functions.

3. Create a Sign Chart (or Number Line): Construct a sign chart (or number line) using the critical points. Mark the critical points on the number line. These points divide the number line into intervals.

4. Test Points in Each Interval: Choose a test value c within each interval created by the critical points. Evaluate the derivative f'(c) at each test point. The sign of f'(c) will tell you whether the function is increasing or decreasing in that entire interval.

   • If f'(c) > 0: The function f is increasing on that interval.
   • If f'(c) < 0: The function f is decreasing on that interval.
   • If f'(c) = 0: While this is unlikely to happen with a random test point, if it does, it might indicate a point of inflection or a local extremum within the interval (though more analysis would be needed to confirm). Typically, you choose a different test point.

5. State the Intervals: Based on the sign chart, identify the intervals where f'(x) > 0. These are the intervals on which the function f is increasing. Express the intervals using interval notation (e.g., (a, b), (-∞, c), [d, e)). Remember to consider whether to include or exclude the endpoints of the intervals, depending on whether the function is defined and continuous at those points and the specific requirements of the problem (sometimes, even if the function is defined at the endpoint, the interval is still defined as open, not closed).

Illustrative Examples

Let's work through some examples to solidify the process:

Example 1: f(x) = x³ - 3x² + 1

1. Find the Derivative:

   • f'(x) = 3x² - 6x

2. Find Critical Points:

   • Set f'(x) = 0: 3x² - 6x = 0
   • Factor out 3x: 3x(x - 2) = 0
   • Solve for x: x = 0 or x = 2
   • The derivative f'(x) = 3x² - 6x is defined for all real numbers, so there are no points where f'(x) is undefined.

3. Create a Sign Chart:

   -----(-)--->(0)----(+)----->(2)----(+)----->
                f'(x)<0    f'(x)>0
   

4. Test Points:

   • Interval (-∞, 0): Choose x = -1. f'(-1) = 3(-1)² - 6(-1) = 3 + 6 = 9 > 0. Therefore, f is increasing on (-∞, 0).
   • Interval (0, 2): Choose x = 1. f'(1) = 3(1)² - 6(1) = 3 - 6 = -3 < 0. Therefore, f is decreasing on (0, 2).
   • Interval (2, ∞): Choose x = 3. f'(3) = 3(3)² - 6(3) = 27 - 18 = 9 > 0. Therefore, f is increasing on (2, ∞).

5. State the Intervals:

   • f is increasing on the intervals (-∞, 0) and (2, ∞).

Example 2: f(x) = x / (x² + 1)

1. Find the Derivative: Use the quotient rule: f'(x) = [(1)(x² + 1) - (x)(2x)] / (x² + 1)² = (1 - x²) / (x² + 1)²

2. Find Critical Points:

   • Set f'(x) = 0: (1 - x²) / (x² + 1)² = 0. A fraction is zero only when its numerator is zero. Therefore, 1 - x² = 0.
   • Solve for x: x² = 1 => x = ±1
   • The derivative f'(x) = (1 - x²) / (x² + 1)² is defined for all real numbers because the denominator (x² + 1)² is always positive.

3. Create a Sign Chart:

   ----(-)--->(-1)----(+)----->(1)----(-)----->
                f'(x)<0    f'(x)>0
   

4. Test Points:

   • Interval (-∞, -1): Choose x = -2. f'(-2) = (1 - (-2)²) / ((-2)² + 1)² = (1 - 4) / (5)² = -3/25 < 0. Therefore, f is decreasing on (-∞, -1).
   • Interval (-1, 1): Choose x = 0. f'(0) = (1 - 0²) / (0² + 1)² = 1/1 = 1 > 0. Therefore, f is increasing on (-1, 1).
   • Interval (1, ∞): Choose x = 2. f'(2) = (1 - (2)²) / ((2)² + 1)² = (1 - 4) / (5)² = -3/25 < 0. Therefore, f is decreasing on (1, ∞).

5. State the Intervals:

   • f is increasing on the interval (-1, 1).

Example 3: f(x) = √(4 - x²)

1. Find the Derivative: Use the chain rule: f(x) = (4 - x²)^(1/2). f'(x) = (1/2)(4 - x²)^(-1/2) * (-2x) = -x / √(4 - x²)

2. Find Critical Points:

   • Set f'(x) = 0: -x / √(4 - x²) = 0. This occurs when the numerator is zero, so x = 0.
   • Points where f'(x) is undefined: The derivative is undefined when the denominator is zero or when the expression inside the square root is negative. √(4 - x²) = 0 when 4 - x² = 0, which means x = ±2. Also, 4 - x² must be greater than or equal to zero (since it's under a square root in the original function), so x must be in the interval [-2, 2]. Therefore, f'(x) is undefined at x = ±2. It is important to remember the domain of the original function.

3. Create a Sign Chart: Remember to consider the domain of f(x), which is [-2, 2].

   ----(+)--->(-2)----(+)---->(0)----(-)----->(2)----(-)--->
                        f'(x)>0    f'(x)<0
   

4. Test Points:

   • Interval (-2, 0): Choose x = -1. f'(-1) = -(-1) / √(4 - (-1)²) = 1 / √3 > 0. Therefore, f is increasing on (-2, 0).
   • Interval (0, 2): Choose x = 1. f'(1) = -(1) / √(4 - (1)²) = -1 / √3 < 0. Therefore, f is decreasing on (0, 2).

5. State the Intervals:

   • f is increasing on the interval (-2, 0).

Important Considerations and Common Pitfalls

• Domain of the Function: Always consider the domain of the original function f(x). The function can only be increasing or decreasing on intervals within its domain. Critical points that fall outside the domain are irrelevant.
• Discontinuities: If the function has any discontinuities (e.g., vertical asymptotes, holes) within the interval you're analyzing, you need to break the interval at the point of discontinuity and analyze each sub-interval separately. The derivative test only applies to continuous functions.
• Critical Points that are Not Extrema: A critical point c where f'(c) = 0 doesn't guarantee a local maximum or minimum. The function could have a horizontal inflection point at c (where the concavity changes, but the function doesn't change from increasing to decreasing or vice versa). To determine the nature of the critical point, you can use the first derivative test (examining the sign change of f'(x) around c) or the second derivative test (evaluating f''(c)).
• Endpoints of Closed Intervals: When dealing with closed intervals [a, b], you need to check the behavior of the function at the endpoints a and b. The derivative test technically applies to open intervals. However, if f'(a) > 0 and f is continuous at a, then f is increasing at a and the interval can be considered closed at a. Similarly, if f'(b) > 0 and f is continuous at b, then f is increasing at b and the interval can be considered closed at b. However, if the derivative is undefined at the endpoint (as in Example 3), or if the function is not continuous, then you typically exclude the endpoint from the interval.

The Second Derivative Test (Brief Mention)

While the first derivative test (analyzing the sign of f'(x)) is the most direct way to find increasing/decreasing intervals, the second derivative test can help classify critical points as local maxima or minima. The second derivative, f''(x), tells us about the concavity of the function:

• If f''(x) > 0, the function is concave up (like a smile).
• If f''(x) < 0, the function is concave down (like a frown).

If f'(c) = 0 (meaning c is a critical point), then:

• If f''(c) > 0, then f(c) is a local minimum.
• If f''(c) < 0, then f(c) is a local maximum.
• If f''(c) = 0, the test is inconclusive.

The second derivative test doesn't directly tell you about intervals of increasing/decreasing behavior, but it helps characterize critical points. You still need the first derivative test to determine the intervals where f is increasing.

Advanced Scenarios

• Piecewise Functions: If f(x) is a piecewise function, you need to analyze each piece separately. Find the derivative of each piece, and then determine the critical points and increasing/decreasing intervals for each piece, being mindful of the domain restrictions for each piece. Pay special attention to the points where the pieces connect; the function must be continuous and differentiable at these points for the derivative test to be valid across those points.
• Implicit Differentiation: If the function is defined implicitly (e.g., by an equation like x² + y² = 1), you'll need to use implicit differentiation to find dy/dx (which is f'(x) in this context). Then, proceed as usual to find critical points and analyze the sign of the derivative.
• Functions with Parameters: Sometimes, the function will contain parameters (constants represented by letters, like a, b, k, etc.). The critical points and increasing/decreasing intervals might depend on the values of these parameters. You'll need to analyze the derivative in terms of these parameters, potentially considering different cases based on the parameter values (e.g., if k > 0, if k < 0, if k = 0).

Conclusion

Determining the intervals where a function is increasing is a fundamental skill in calculus. By finding the derivative, identifying critical points, and using a sign chart, you can systematically analyze the function's behavior and pinpoint the intervals where it's increasing. Remember to consider the domain of the function, potential discontinuities, and the nuances of endpoints when stating your final answer. Mastering this process provides a strong foundation for understanding the behavior of functions and solving a wide range of calculus problems.