3. APPLICATIONS

1) Find the intervals of concavity and convexity of the function f(x)=ln(x²+1)

First, complete the following in your exercise book:

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

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

• Write down the intervals when the function is concave or convex. Where are the points of inflexion?

Now look carefully at the graph of y=f''(x) in the window

• Where does f''(x) cut the X-axis?

• What is the sign of f''(x) on either side of these points?

The graph of y=f(x) appears when you change the value of x. Look at its behaviour and check your results.

2) Find the intervals of concavity and convexity and the points of inflexion of the function f(x)=x-1/x

• Find f''(x) and check that f''(x)=0 does not have a solution, but that its sign changes depending on whether x<0 or x>0

• Write down the intervals of concavity and convexity

To check your results make the value of the SECOND DERIVATIVE equal to 1 in the top part of the window. The graph of f''(x) will be drawn. Note how it does not actually cut the X-axis.

• In this case what happens at x=0? What is the behaviour of f''(x) on either side of 0.

• Find f''(x) and solve the equation: f''(1)=0 

• What is the value of f'''(1) for the value of a obtained?

Drag the red point with the mouse until it is on the X-axis or change the value of f´´(1) en the top part of the window until it is equal to 0.

• Now f''(1)=0. What is the value of a?

The graph of will be drawn as you change the value of x. Thus you can check for a POINT OF INFLEXION at x=1.