In this section, we will see theorems that help us determine whether a function is increasing, decreasing or constant on an interval.

Theorem

A function f is said to be increasing on the interval I if for each , wherever .

It is decreasing if wherever .

NB

The function f is said to be monotonic on an interval I if it is increasing, decreasing or constant throughout this interval.

Theorem

If  is continuous on an interval I and if for each  in the interior of I, then  is increasing on I.

Similarly, if  is continuous on I and for all interior points x in I, then f(x) is decreasing on I.

Now to find the increase and the decrease intervals of the function, we have to study the sign of its first derivative. In many cases, that may be difficult. The next theorem helps us to study the sign of the derivative at a whole interval by finding its sign at only one point.

Theorem

Let f be a function, continuous on an interval I. If  at all points , the function is either positive all over I or negative all over I.

NB

The theorem assumes that the function is continuous, and so it exists, at every point in I.

Now, if the function is positive at any point in I, the function will be positive throughout the interval. The same applies if it is negative. That can be used in finding the sign of  on a whole interval by substituting only once. That is shown in the example.

Definition

A critical point of a function f(x) is a point where  or  does not exist.

Therefore, if as long as the derivative of the function is continuous, there lie between the critical points of the function intervals satisfying the requirements of the last theorem.

Example

Study the monotonicity of the following function on R:

Solution