Find absolute extrema
To find the absolute extrema of a function, we need to consider the maximum and minimum values over the entire domain of the function
To find the absolute extrema of a function, we need to consider the maximum and minimum values over the entire domain of the function. This means that we need to look at the highest and lowest points the function reaches.
To find the absolute extrema, follow these steps:
1. Determine the domain of the function. This is the set of all possible x-values for which the function is defined.
2. Find the critical points of the function. These are the points where the derivative of the function is either zero or undefined. To do this, take the derivative of the function, set it equal to zero, and solve for the x-values that satisfy this equation.
3. Evaluate the function at the critical points and endpoints of the domain. Plug the x-values from step 2 and the endpoints of the domain into the original function. This will give you the corresponding y-values or function values.
4. Compare the y-values obtained in step 3 to determine the absolute extrema. The largest y-value will represent the absolute maximum, and the smallest y-value will represent the absolute minimum.
Note: If the function is continuous over a closed interval (i.e., defined for all values between two endpoints), you can also use the First Derivative Test and Second Derivative Test to verify if the critical points are relative maxima or minima.
Let’s go through an example:
Consider the function f(x) = x^3 – 3x^2 – 36x + 2 on the interval [-5, 5].
1. The domain of the function is [-5, 5].
2. Find the critical points by taking the derivative of the function and setting it equal to zero:
f'(x) = 3x^2 – 6x – 36
Setting this equal to zero:
3x^2 – 6x – 36 = 0
Solving the quadratic equation, we get:
x = -3, 6
3. Evaluate the function at the critical points and endpoints:
f(-5) = (-5)^3 – 3(-5)^2 – 36(-5) + 2 = 172
f(-3) = (-3)^3 – 3(-3)^2 – 36(-3) + 2 = -114
f(5) = (5)^3 – 3(5)^2 – 36(5) + 2 = -18
4. Compare the y-values:
The maximum value is 172, which occurs at x = -5, and the minimum value is -114, which occurs at x = -3.
Therefore, the absolute maximum is f(-5) = 172, and the absolute minimum is f(-3) = -114.
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