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Clicker Question 1 What are the critical numbers of f (x ) = |x + 5| ? A. 0 B. 5 C. -5 D. 1/5 E. It has no critical numbers

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Clicker Question 2 What are the critical numbers of g (x ) = x 2 e 4x ? A. 0 and -2 B. -1/2 C. 0 and 1/2 D. 0 and -1/2 E. It has no critical numbers.

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Finding and Classifying Extrema (4/1/09 – no fooling) In many applications, we are concerned with finding points in the domain of a function at which the function takes on extreme values, either local or global. To find these, we look at: Critical points Endpoints the domain, if they exist Behavior as the input variable goes to ∞

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Classifying critical points: The Value Test If the values of a function just to the left and right of a critical point are below the value at the critical point, then that point represents a local maximum of the function. Similarly for local minimum.

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Classifying critical points: The First Derivative Test If the values of the first derivative are positive to the left but negative to the right of a critical point, the value of the function at that point is a local maximum. (Why??) Similarly for local minimum.

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Classifying critical points: The Second Derivative Test If the value of the second derivative is negative at a critical point, then the value of the function at that point is a local maximum. (Why??) Similarly for local minimum. You should use whichever of these tests is simplest for the given problem.

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Clicker Question 3 Suppose f is a function which has critical numbers at x = 0, 3, and 6, and suppose f '(2) = -1 and f '(4) = 1.5. Then at x = 3, f definitely has a A. local maximum B. local minimum C. global maximum D. global minimum E. neither a max nor a min

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Clicker Question 4 Suppose f is a function which has critical numbers at x = 0, 3, and 6, and suppose f ''(3) = -2. Then at x = 3, f definitely has a A. local maximum B. local minimum C. global maximum D. global minimum E. neither a max nor a min

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Inflection Points A point on a function at which the concavity changes is called an inflection point of the function These usually occur at points where the second derivative is 0 or does not exists, but not necessarily (e.g., f (x) = x 4 at 0).

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Assignment for Friday Read Section 4.3. In Section 4.3, do Exercises 1, 3, 5, 11, 15, 17, 25, 33, 44, 45, 66. On Friday, HW#3 will be given out and will be due Tuesday April 7 at 4:45 pm.

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