5.1 Day 2 The Candidate Test / Finding Absolute Extrema on a closed interval

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5.1 Day 2 The Candidate Test / Finding Absolute Extrema on a closed interval

Review Critical Numbers: x – values at which the derivative is zero or not differentiable. A relative maxima/minima occurs at a critical number; but not all critical numbers produce relative extrema (think of functions with vertical asymptotes).

Finding Absolute Extrema on a Closed Interval A.K.A The Candidates Test (The function must be continuous on the closed interval.) Guidelines: 1. Find all critical numbers of f in (a, b). 2. Evaluate f at each critical number in (a, b). 3. Evaluate f at each endpoint. 4. The least of these is the minimum. The greatest is the maximum. Extrema Extra

Example 1 Locate the absolute extrema of the function on the closed interval.

Example 2 Same directions.

#25 from the Homework Same directions.

Example 3 Sketch a function on the interval [-1, 4] with the following characteristics: Relative Minimum at x = 0. Absolute Minimum at x = -1. Absolute Maximum at x = 3.

Last ones of the day…. True / False (homework questions 55 – 56) 55. The maximum of a function that is continuous on a closed interval can occur at two different values in an interval. 56. If a function is continuous on a closed interval, then it must have a minimum on the interval. HW: 5.1 Day 2 (skip #25, 55, 56)