How do you determine the absolute maximum and minimum values of the function f(x) = 2 – x² over the interval [-1, 1] and where these extrema are located?

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To find the absolute maximum and minimum values of the function f(x) = 2 – x² over the interval [-1, 1], we can follow a systematic approach that includes checking the endpoints of the interval as well as any critical points within the interval.

Step 1: Identify the Critical Points

A critical point occurs where the derivative of the function is zero or undefined. First, we need to find the derivative of f(x):

f'(x) = -2x

Next, we set the derivative equal to zero to find the critical points:

-2x = 0

Solving this gives:

x = 0

Step 2: Evaluate the Function at the Endpoints and Critical Points

Now, we will evaluate the function at the endpoints x = -1 and x = 1, as well as the critical point x = 0:

  • At x = -1: 
    f(-1) = 2 - (-1)² = 2 - 1 = 1
  • At x = 0: 
    f(0) = 2 - (0)² = 2 - 0 = 2
  • At x = 1: 
    f(1) = 2 - (1)² = 2 - 1 = 1

Step 3: Compare the Values

We have the following function values:

  • f(-1) = 1
  • f(0) = 2
  • f(1) = 1

Conclusion

From our evaluations, we can see that:

  • The absolute maximum value of f(x) over the interval [-1, 1] is 2, occurring at x = 0.
  • The absolute minimum value of f(x) over the interval [-1, 1] is 1, occurring at both x = -1 and x = 1.

In conclusion, the absolute max and min values of the function f(x) = 2 – x² on the interval [-1, 1] are found, along with their locations:

  • Maximum: 2 at x = 0
  • Minimum: 1 at x = -1 and x = 1

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