Find absolute extrema
Use critical values and END POINTS in the function
The absolute extrema of a function are the highest and lowest values of the function in a given interval. To find the absolute extrema of a function, follow these steps:
1. Identify the interval: Determine the interval over which you want to find the absolute extrema. This could be a closed interval [a,b] or an open interval (a,b).
2. Find the critical points: Find the values of x where the derivative of the function is equal to zero or undefined. These are the critical points of the function.
3. Check the endpoints: Evaluate the function at the endpoints of the interval.
4. Compare values: Compare the values of the function at the critical points and endpoints, and identify the highest and lowest values.
5. State the answer: State the highest and lowest values as the absolute maximum and minimum values, respectively.
Example: Find the absolute extrema of the function f(x) = x^3 – 3x^2 + 1 on the interval [-2,2].
1. Interval: [-2,2]
2. Critical points: Take the derivative of the function to find the critical points
f'(x) = 3x^2 – 6x
Setting f'(x) = 0 yields the values x = 0 and x = 2.
3. Endpoints: Evaluate the function at the endpoints
f(-2) = (-2)^3 – 3(-2)^2 + 1 = -3
f(2) = 2^3 – 3(2)^2 + 1 = -3
4. Compare values: Compare the values of the function at the critical points and endpoints
f(0) = 1
f(2) = -3
Thus, the absolute maximum value is 1 at x=0 and the absolute minimum value is -3 at x=2 and x=-2.
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