2. FINDING THE LOCAL MAXIMA AND MINIMA OF A FUNCTION

In the window you can see the curves of the function
f(x)=x⁴-2x²+1, its derivative  f'(x)=4x³-4x 

and its second derivative f''(x)=12x²-4

In order to find the local extremes we must:

• Solve the equation: f'(x)=4x³-4x=0  

        (Solutions: x=-1, x=0, x=-1)

• Find the sign of the second derivative for these values

x=-1, f'(x)=0, f''(x)>0  minimum at (-1,-2)

x=0, f'(x)=0, f''(x)<0  maximum at (0,-1)

x=1, f'(x) = 0, f''(x)>0  minimum at (1,-2)

In the window you can see the curves of the function

y=2x/(x²+1) and its derivative y=f'(x)

• Look at these curves. Where does y=f'(x) cut the X-axis? What is the sign of f'' for each of these x values?

• Find the derivative: f'(x), solve f'(x)=0 and check that the solutions are x=1 and x=-1

Change the value of x and you will see another curve appear, which is the graph of  f''(x)

• What are the signs of f''(1) and f''(-1)?

x=-1, f'(x)=0, f''(x)>0  minimum at (-1,1)

x=1, f'(x)= 0, f''(x)<0  maximum at (1,1)