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This guide covers the process of finding critical numbers and extrema in continuous functions on a closed interval. It emphasizes the importance of calculating the derivative, setting it to zero, and solving for critical values. The method includes testing intervals to ensure valid critical numbers. An example illustrates how to determine the absolute maximum and minimum values on a specific interval, showcasing the steps taken to find these extrema comprehensively.
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Critical Numbers Example 2: Take the derivative of f(x) Set the derivative equal to zero Solve for x *Remember to test the regions to make sure the critical numbers are valid Example 1: you can’t take the square root of a negative number, so the only critical number is 0
Finding Extrema(closed interval, continuous function) • Find critical numbers • Find endpoint values • Find the function values for the x values determined in step one and two • Identify extrema • Largest overall function value is the absolute max • Smallest overall function value is the absolute min
Finding Extrema Example Find the absolute min and max for the function on the interval [-3,5] : Absolute max is 66 and occurs at x=5 and the absolute min is -15 and occurs at x=2
