4095: Find the absolute maximum value

Question

Find the absolute maximum value for the function f(x) = x² - 4, on the interval [-3, 0) U

• -3
• 5
• 2
• No maximum exists because f is not continuous

Answer 1

To find the absolute maximum, we need to find the local extremas using the first derivative test and evaluate f(x) at the bounds.  Then compare the maximum values to see which one is the highest value within the bound.

 

 

The first derivative test means to set f(x) equal to zero.

 

f'(x) = 0

 

2x = 0

 

x = 0

 

 

We can either evaluate the derivative of f(x) at x=-1 and at x=1 to see if the derivative changes from positive to negative, or we can recognize that positive parabolas only have a local minimum.

 

Because f(0) is a local minimum, we only need to evaluate f(x) at the bounds.

 

f(-3) = (-3)² - 4

        = 9 - 4

        = 5

 

f(0) = 0² - 4

       = -4

 

Because f(0) is not included in the bounds, f(x) is close to -4 when x is close to zero.

 

After comparing the two f(x) values, the absolute maximum is 5.