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In this article, we will study the concept of increasing and decreasing functions, their properties, graphical representation, and theorems to test for increasing and decreasing functions along with examples for a better understanding.
Increasing Functions. A function is "increasing" when the y-value increases as the x-value increases, like this: It is easy to see that y=f (x) tends to go up as it goes along.
We are now learning that functions can switch from increasing to decreasing (and vice--versa) at critical points. This new understanding of increasing and decreasing creates a great method of determining whether a critical point corresponds to a maximum, minimum, or neither.
This video explains how to use the first derivative and a sign chart to determine the intervals where the function is increasing and decreasing and how to express the answer using interval...
Increasing and decreasing are properties in real analysis that give a sense of the behavior of functions over certain intervals. For differentiable functions, if the derivative of a function is positive on an interval, then it is known to be increasing while the opposite is true if the function's derivative is negative.
Definition of an Increasing and Decreasing Function. Let y = f (x) be a differentiable function on an interval (a, b). If for any two points x1, x2 ∈ (a, b) such that x1 < x2, there holds the inequality f(x1) ≤ f(x2), the function is called increasing (or non-decreasing) in this interval. Figure 1.
What are increasing and decreasing functions? A function f (x) is increasing on an interval [a, b] if f' (x) ≥ 0 for all values of x such that a< x < b. If f' (x) > 0 for all x values in the interval then the function is said to be strictly increasing.