Chapter 7 Differential Calculus

Section 7.4 Properties of Functions

7.4.2 Monotony

The derivative of a function can be used to study the growth behaviour, i.e. whether the function values increase or decrease for increasing values of x. For this purpose, we consider a function f:D that is differentiable on ]a;b[ D:

If f'(x)0 for all x between a and b, then f is monotonically decreasing on the interval ]a;b[ .
If f'(x)0 for all x between a and b, then f is monotonically increasing on the interval ]a;b[ .
Thus, it is sufficient to determine the sign of the derivative f' to decide whether a function is monotonically increasing or decreasing on the interval ]a;b[ .
Example 7.4.1
The function f:,x x3 is differentiable with f'(x)=3 x2 . Since x2 0 for all x, we have f'(x)0, and therefore f is monotonically increasing.
For g: with g(x)=2 x3 +6 x2 -18x+10, the function g'(x)=6 x2 +12x-18=6(x+3)(x-1) has the roots x1 =-3 and x2 =1. If the monotony of the function g is investigated, then three regions are to be distinguished in which g' has a different sign.
The following table is used to determine in which region the derivative of g is positive or negative. These regions correspond to the monotony regions of g. The entry " +" says that the considered term is positive on the given interval. If the term is negative, then " -" is entered.

xx<-3-3<x<11<x x+3-++ x-1--+ g'(x)+-+ g   is monotonically      increasing      decreasing      increasing   

For the function h:{0} with h(x)= 1 x , we have h'(x)=- 1 x2 , that is h'(x)<0 for all x0.
Even though the function h exhibits the same monotony behaviour for the two subregions x<0 and x>0, it is not monotonically decreasing on the entire region. As a counterexample let us consider the function values h(-2)=- 1 2 and h(1)=1. Here, we have -2<1 but also h(-2)<h(1). This corresponds to an increasing growth behaviour if we change from one subregion to the other. The statement that the function h is monotonically decreasing on ]-;0[ thus means that the restriction of h on this interval is monotonically decreasing. Moreover, the function h is also monotonically decreasing for all x>0.