INCREASE AND DECREASE : Calculating the intervals
Analysis
 

2. CALCULATING INTERVALS OF INCREASE OR DECREASE
EXAMPLE 1

In this window you can see the curve of the function


 f(x)=x3-3x+1 and its derivative f'(x)=3x2-3

Change the value of x and check whether the function is increasing or decreasing by looking at the sign of the derivative when x=-1.5   x=0   x=2

In order to find the intervals in which the function increases or decreases we can:

  • Solve the equation: f'(x)=0 

          (Solutions: x=1, x=-1)

  • Find the sign of the derivative on either side of these values

x<-1, f'(x)>0, f increases on (-¥,-1)

-1<x<1, f'(x)<0, f decreases on (-1,1)

x>1, f'(x) > 0, f increases on (1,+¥)

 
EXAMPLE 2

The window shows the derivative y=f'(x) of the function y=x+1/x

Look carefully at the sign of the derivative in the window. Where does the derivative of the function cut the X-axis? What happens when x=0?

  • Find f´ and solve the equation f'(x)=0

  • Check that the solutions are:  x=1, x=-1

Change the value of x and the graph of y=f(x) will appear. Observe its behaviour.

x<-1, f'(x)>0, f increases on (-¥,-1)

-1<x<0, f'(x)<0, f decreases on (-1,0)

0<x<1, f'(x)<0, f decreases on (0,1)

x>1, f'(x)>0, f increases on (1,+¥)
       
           
  María José García Cebrian
 
Spanish Ministry of Education. Year 2001
 
 

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