2. FINDING INTERVALS OF CONCAVITY-CONVEXITY AND POINTS OF INFLEXION

EXAMPLE 1

In the window you can see the graphs of the function f(x)=4xe^-x and of its second derivative f''(x)=(4x-8)e^-x

In order to find the intervals of concavity and convexity follow this procedure:

• Solve the equation: f''(x)=(4x-8)e^-x=0

• Solutions: x=2

• Find the sign of the second derivative on either side of this value.

x<2, f''(x)<0  concave on the interval (-¥,2)

x=2, f''(x)=0  point of inflexion at (2,1.08)

x>2, f''(x)>0  convex on the interval (2,+¥)

EXAMPLE 2

Now you can see the graph of the function y=x/(x²-1) and its second derivative y=f''(x) in the window.

• Find the derivative: f''(x), solve the equation: f''(x)=0 and check that the solution is:  x=0

• Find the sign of f''(x) on either side of 0

Look carefully at the graphs in the window and change the value of x to check your results.

• What happens on either side of x=-1 and x=1?

• Note that as they do not belong to the domain of the function that they are not points of inflexion, even though the direction of the curve changes.

x<-1, f''(x)<0   concave on the interval (-¥,-1)

-1<x<0, f''(x)>0  convex on the interval (-1,0)

x=0 f''(0)=0 point of inflexion at (0,0)

0<x<1, f''(x)<0   concave on the interval (0,1)

x>1 f''(x)>0  convex on the interval (1,+¥)