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Increasing and Decreasing Functions

A function f is described as increasing on an interval I if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2). A function f is described as decreasing on an interval I if for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2). It is often useful to know where the graph of a differentiable function increases (rises from left to right) and where it decreases (falls from left to right) over an interval.

Increasing/Decreasing Test

Suppose that f is continuous on [a, b] and differentiable on (a, b).

  1. If f '(x) > 0 on an interval, then f is increasing on that interval.
  2. If f '(x) < 0 on an interval, then f is decreasing on that interval.
  3. If f '(x) = 0 on an interval, then f is constant on that interval and neither increases nor decreases.

View the presentation on determining where a function is increasing and decreasing.

Monotonicity

A function that is consistently increasing or consistently decreasing on an interval is said to be monotonic on the interval. View the presentation on monotonicity.

Relating the Behavior of f(x) to the Sign of f'(x)

It is possible to deduce information about the behavior of the graph of f(x) from the sign of its derivative. Because f'(x) represents the slope of the curve y = f(x) at the point (x, f(x)), we can determine the direction of the curve as the function takes on each value of x.

View the presentation investigating the relationship of the graph of a function to the graph of its derivative.

First Derivative Test

Suppose that c is a critical number of a continuous function f, and that f is differentiable at every point in some interval containing c except possibly at c itself.

  1. If f ' changes from positive to negative at c, then f has a local maximum at c.
  2. If f ' changes from negative to positive at c, then f has a local minimum at c.
  3. If f ' does not change sign at c, then f has no local maximum or minimum at c.

View the presentation on using the First Derivative Test to find relative extrema.

Increasing and Decreasing Functions Practice

Increasing and Decreasing Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of increasing and decreasing functions. 

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