Increasing and Decreasing Functions

In sketching the graph of a function, it is very useful to know where it raises and where it falls. The graph shown below raises from A to B, falls from B to C, and raises again from C to D.

The function f is said to be increasing on the interval [a,b], decreasing on [b,c], and increasing again on [c,d].

A function f is called increasing on an interval I if f(x1) ≤ f(x2) whenever x1 < x2 in I. It is called decreasing on I if f(x1) ≥ f(x2) whenever x1 < x2 in I. A function that is completely increasing or completely decreasing on I is called monotonic on I.

Usually, by looking at the graph of the function you can say whether the function is increasing or decreasing or neither. The graph of an increasing function does not fall as you go from left to right while the graph of a decreasing function does not rise as you go from left to right. But if you are not given the graph, how do you decide whether a given function is monotonic or not?

Theorem 1

Let I be an open interval. Let f:I→R be differentiable. Then

  1. f is increasing if and only if f′(x) ≥ 0 for all x in I.
  2. f is decreasing if and only if f′(x) ≤0 for all x in I.

Geometrical Interpretation

If on an interval I = [a,b] a function f(x) increases, then the tangent to the curve y = f(x) at each point on this interval forms an acute angle ϕ with the x-axis or (at certain points) is horizontal. The tangent of this angle is not negative. Therefore f′(x) = tan ϕ ≥ 0.

If the function f(x) decreases on the interval [b,c] then the angle of inclination of the tangents form an obtuse angle. The tangent of this angle is not positive. Therefore, f′(x) = tan ψ ≤ 0.

Theorem 2

Let f′ be positive on I. Then f is strictly increasing on I.

Let f′ be negative on I. Then f is strictly decreasing on I.