Problem 50 - Calculus

Question

True or false? Give an explanation for your answer. A function which is monotonic on an interval is either increasing or decreasing on the interval.

Step-by-Step Solution

Answer

True, a monotonic function is either increasing or decreasing on the interval.

Step 1: Define Monotonic Function

A function is said to be monotonic if it is either entirely non-increasing or non-decreasing on a given interval. This means that the function doesn't change its direction of growth within that interval.

Step 2: Understanding Increasing and Decreasing

A function is increasing on an interval if for any two points, the function's value at the second point is greater than or equal to the first. Conversely, a function is decreasing if the value at the second point is less than or equal to that at the first point for any two points in the interval.

Step 3: Analyze the Statement

Given that a function is monotonic, it implies that the function is either increasing or decreasing. By definition, a monotonic function does not switch between increasing and decreasing within the interval. Therefore, a monotonic function on an interval can only be classified as either increasing or decreasing.

Step 4: Conclusion

Therefore, the statement is true. A function that is monotonic over an interval does not change its trend and is either increasing or decreasing throughout the interval.

Key Concepts

Increasing Function

To grasp the concept of an increasing function, let's imagine you're climbing up a steady hill. In mathematical terms, this scenario represents an increasing function. When we say that a function is increasing on an interval, it means that for any two points within that interval, if the first point is less than the second, then the function value at the first point is also less than or equal to the function value at the second point.
For example, consider the function \( f(x) = x^2 \) over the interval \( [0, \, \infty) \). Here, as \( x \) moves from left to right (from smaller to larger values), \( f(x) \) gets larger, thus showing an increasing trend.

• In simpler terms, the output (or function value) "increases" as the input gets larger within the interval.
• When graphed, an increasing function looks like it is rising or staying flat as you move from left to right.

This property is crucial in understanding functions and their behavior, especially within specific intervals.

Decreasing Function

A decreasing function provides an opposite experience to that of an increasing function. Imagine sliding down a gentle slope; this is how a decreasing function behaves. If a function is decreasing over an interval, for any two points therein, if the first point is less than the second, then the function's value at the first point is greater than or equal to the value at the second.
Let's take \( f(x) = -x \) as an example over the interval \( (-\infty, \, \infty) \). As \( x \) becomes larger, \( f(x) \) becomes smaller, indicating a decreasing trend.

• In simple terms, as you input larger values into the function, the output values shrink or stay the same.
• Graphically, a decreasing function will have a downward or level trend from left to right.

Understanding when a function is decreasing helps in analyzing trends and predicting behavior over certain intervals.

Function Behavior on Interval

When discussing function behavior on an interval, we are essentially talking about how the function behaves as we pick different values over that specified section of the domain. A function is classified as monotonic on an interval if it is either consistently increasing or consistently decreasing throughout that interval.
This characteristic is vital because it simplifies the analysis of a function's behavior without any interruptions in its growth pattern.

• A monotonic function will not have any fluctuations up and down within the interval.
• The behavior of being either increasing or decreasing is observed in its continuity and constant slope over the interval.

Thus, if a function is monotonic over an interval, it either steadily climbs or descends, ensuring that trend lines are predictable and interpretable for both mathematical analysis and real-world applications.