Test Corrections On a separate sheet (not the original test), correct all your mistakes carefully. Hand both in, not stapled together. Do this work on your own or with help from me only. Honor code! You’ll get back 1/3 of the lost points which are corrected. Due Wednesday Feb 29 at class time.

Clicker Question 1 What are the absolute maximum and minimum values of f (x ) = 2x + (1/x 2 ) on the interval [1/2, 2]? A. 5 and 3 B and 3 C. 5 and 4.25 D. ½ and 1 E. 2 and ½

Finding and Classifying Extrema (2/27/12) 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  ∞

Classifying critical points: The Value Test If the values (“values” means “output 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. Note that this method requires you to compute three output values.

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. Similarly for local minimum. Note that here you need only determine the sign of the derivative’s value at two points

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. Similarly for local minimum. Note that this requires finding only the sign of the value of the second derivative at one point. But the downside is: You have to compute the second derivative

Clicker Question 2 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

Clicker Question 3 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

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).

Assignment for Wednesday In Section 4.3, do Exercises 1, 3, 5, 9, 11, 15, 17, and 25. Test correction are due Wednesday at class time.