5.3 Determining Intervals on Which a Function is Increasing or Decreasing

4 min read•january 29, 2023

ethan_bilderbeek

Sumi Vora

ethan_bilderbeek

Sumi Vora

AP Calculus AB/BC ♾️

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Determining Intervals on Which a Function is Increasing or Decreasing

In order to determine the intervals on which a function is increasing or decreasing, we first need to understand the concept of the derivative. The derivative of a function is a measure of how the function is changing at a given point. If the derivative is positive at a certain point, this means that the function is increasing at that point. If the derivative is negative at a certain point, this means that the function is decreasing at that point.

One way to determine the intervals on which a function is increasing or decreasing is by using the First Derivative Test. The First Derivative Test states that if the derivative of a function changes from positive to negative at a point, then the function has a local maximum at that point. If the derivative changes from negative to positive at a point, then the function has a local minimum at that point.

For example, consider the function f(x) = x^2. The derivative of this function is f'(x) = 2x. At x = 0, the derivative is equal to 0, which means that the function is neither increasing nor decreasing at that point. However, as we move away from x = 0 in either direction, the derivative becomes positive, indicating that the function is increasing on the entire interval (-infinity, infinity).

Another way to determine the intervals on which a function is increasing or decreasing is by using the Second Derivative Test. The Second Derivative Test states that if the second derivative of a function is positive at a certain point, then the function has a local minimum at that point. If the second derivative is negative at a certain point, then the function has a local maximum at that point.

For example, consider the function f(x) = x^3. The second derivative of this function is f''(x) = 6x. At x = 0, the second derivative is equal to 0, which means that the function is neither concave up nor concave down at that point. However, as we move away from x = 0 in either direction, the second derivative becomes positive, indicating that the function is concave up on the entire interval (-infinity, infinity)

Examples:

For the function f(x) = x^2 + 1, the first derivative is f'(x) = 2x and the second derivative is f''(x) = 2. The first derivative is positive for all x, indicating that the function is increasing on the entire interval (-infinity, infinity). The second derivative is positive for all x, indicating that the function is concave up on the entire interval (-infinity, infinity)
For the function f(x) = x^3, the first derivative is f'(x) = 3x^2 and the second derivative is f''(x) = 6x. The first derivative is positive for all x except for x=0, indicating that the function is increasing on the intervals (-infinity,0) and (0,infinity). The second derivative is positive for all x, indicating that the function is concave up on the entire interval (-infinity, infinity)
For the function f(x) = -x^2, the first derivative is f'(x) = -2x and the second derivative is f''(x) = -2. The first derivative is negative for all x, indicating that the function is decreasing on the entire interval (-infinity, infinity). The second derivative is negative for all x, indicating that the function is concave down on the entire interval (-infinity, infinity)
For the function f(x) = sin(x), the first derivative is f'(x) = cos(x) and the second derivative is f''(x) = -sin(x). The first derivative changes sign periodically, indicating that the function has local extrema (maxima and minima) at certain points. To determine where these extrema occur, we set the derivative equal to 0 and solve for x. For example, if we set f'(x) = 0, we get cos(x) = 0, which occurs when x = pi/2, 3pi/2, etc. These values correspond to local maxima of the function. The second derivative is negative for all x, indicating that the function is concave down on the entire interval (-infinity, infinity).

In conclusion, determining the intervals on which a function is increasing or decreasing involves finding the first and second derivatives of the function and analyzing their behavior. The First Derivative Test allows us to identify local maxima and minima by analyzing the sign change of the first derivative, while the Second Derivative Test allows us to identify the concavity of the function by analyzing the sign of the second derivative. By combining these two techniques, we can get a complete picture of the behavior of a function and make informed conclusions about its properties.

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👑Unit 1 – Limits & Continuity

🤓Unit 2 – Fundamentals of Differentiation

🤙🏽Unit 3 – Composite, Implicit, & Inverse Functions

👀Unit 4 – Contextual Applications of Differentiation

✨Unit 5 – Analytical Applications of Differentiation

5.0Unit 5 Overview: Analytical Applications of Differentiation

5.1Using the Mean Value Theorem

5.2Extreme Value Theorem, Global vs Local Extrema, and Critical Points

5.3Determining Intervals on Which a Function is Increasing or Decreasing

5.4Using the First Derivative Test to Determine Relative (Local) Extrema

5.5Using the Candidates Test to Determine Absolute (Global) Extrema