3. APPLICATIONS

1) Find the local extremes of the function

f(x)=(x²-3)e^x-1

Do the following in your exercise book:

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

• Find f''(x) and its sign for these values

Now look at the curves of f'(x) and f''(x) in the window.

• Where does f'(x) cut the X-axis?

• What is the behaviour of f''(x) at these points?

2) Find the local extremes of the function

f(x)=x³/(x²-3)

• You now know the procedure you need to follow. Solve the equation: f'(x)=0. Then find f''(x) and its sign for the x values you obtain. In this example, what happens when x=0?

Make the value of the DERIVATIVE equal to 1 in the top part of the window. The curves of f'(x) and f''(x) will appear.

• Note that the sign of f' is the same on both sides of zero. There is no maximum or minimum of the function when x=0. Later on we will see what to do in these cases.

3) Find the value of a so that f(x)=x³+ax has a local extreme at x=1. Is it a maximum or minimum?

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

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

• Is there a maximum or minimum at x=1?

Look at the curve of f' in the window. Change the value of f'(1) until the red point is on the X-axis. You can also do this by dragging the point with the mouse.

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

Change the value of x and the curve of y=f(x) will appear. You can use this to check that there is a maximum or minimum at x=1.