Step-by-Step Guide to Finding Absolute Extrema of a Function Over a Given Interval

Candidates Test (absolute extrem. a)

To find the absolute extrema of a function over a given interval, we need to identify the highest and lowest points on the graph

To find the absolute extrema of a function over a given interval, we need to identify the highest and lowest points on the graph. Here’s how you can approach this type of problem step by step:

1. Identify the domain of the function: The domain is the set of all possible input values for the function. Make sure the given interval is within the domain. For example, if the function is defined for all real numbers, the domain would be (-∞, ∞).

2. Take the derivative of the function: The derivative gives us information about the slope of the function at different points. To find the relative extrema, we need to identify where the derivative is equal to zero or does not exist. These points will potentially be the candidates for the absolute extrema.

3. Solve for critical points: Set the derivative equal to zero and solve for x to find the critical points. The critical points are where the function could potentially have a maximum or minimum value.

4. Check the endpoints of the interval: Evaluate the function at the endpoints of the interval to see if any of them could be absolute extrema. This step is necessary because the endpoints are not critical points and may contain the maximum or minimum value.

5. Find the function values at critical points and endpoints: Plug the critical points and endpoints back into the original function to get the corresponding function values.

6. Compare the function values: Compare the function values obtained from step 5 to determine the highest and lowest values, which are the absolute extrema.

Let’s illustrate this process with an example:

Example: Find the absolute extrema of the function f(x) = x^2 – 4x + 3 on the interval [0, 4].

Step 1: The domain of the function is all real numbers because there are no restrictions on the input, so the interval [0, 4] is within the domain.

Step 2: Take the derivative of the function f(x) = x^2 – 4x + 3:
f'(x) = 2x – 4

Step 3: Set the derivative equal to zero and solve for x:
2x – 4 = 0
2x = 4
x = 2

So, x = 2 is a critical point.

Step 4: Check the endpoints of the interval:
f(0) = 0^2 – 4(0) + 3 = 3
f(4) = 4^2 – 4(4) + 3 = 3

Step 5: Evaluate the function at the critical point:
f(2) = 2^2 – 4(2) + 3 = -1

Step 6: Compare the function values:
The function values we obtained are:
f(0) = 3
f(2) = -1
f(4) = 3

The highest function value is 3, which occurs at both x = 0 and x = 4. The lowest function value is -1, which occurs at x = 2.

Therefore, the absolute maximum value is 3, and it occurs at x = 0 and x = 4. The absolute minimum value is -1, and it occurs at x = 2.

That’s it! You’ve successfully found the absolute extrema of the given function over the given interval.

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