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Local Vs Global Max/Min Local Max/Min A local maximum/minimum is a point (p, f(p)) on the graph of that has the largest/smallest function value around. Formally we say there is a local max/min at x = p if f(p) is the largest/smallest function value in a neighborhood or set of points around p. Local maxima and minima occur at points where f x 0 or where f x is undefined. It’s important to remember that not all points where f x 0 or is undefined are local extrema but that all local extrema are at points where f x 0 or is undefined. The requirement that f x 0 or is undefined is known as a necessary but not sufficient condition for local extrema. It also gives us a good way to look for them. a.) Find f '(x) b.) Find all points where f '(x) = 0 or is undefined. These are known as the critical points (CPs) of f(x). c.) Test each CP to see if it is a local max/min or neither. d.) The first derivative test says that if f '(x) changes sign at a CP then it is a local max/min. If f '(x) changes from positive to negative at the CP then the CP is a local max, if f '(x) changes from negative to positive at the CP then the CP is a local min and if f '(x) doesn’t change sign at the CP then the CP is neither a local max nor a local min. e.) The second derivative test says that if f "(x) is positive at the CP then the CP is a local min, if f "(x) is negative at the CP then the CP is a local max and if f "(x) is zero at the CP the second derivative test is inconclusive. x5 16 x3 Example: f x 5 3 Find the CPs: f x x 4 16 x 2 x 2 x 2 16 x 2 x 4 x 4 0 when x 0 or x 4 or x 4 Test the CPs: Using the first derivative test, when x = 4 f x 0 for values to the left of 4 and f x 0 for values to the right of 4 (try evaluating f 3.9 and f 4.1 to see what happens). This tells us that there is a local minimum at x = 4. Using the second derivative test, when x = -4 f x 4x3 32x and f 4 4 4 32 4 256 128 128 0 . 3 This tells us that there is a local maximum at x = -4. When x = 0 the second derivative test is inconclusive because f 0 0 . Using the first derivative test we see that f x 0 both to the right of 0 and to the left of 0. (try evaluating f .1 and f .1 to see what happens) This tells us that x = 0 is neither a local max nor a local min. Global Max/Min A global max/min is a point (p, f(p)) on the graph of f(x) that has the largest/smallest function value on the domain of f(x). Formally we say there is a global max at x = p if f p f x for all x in the domain and a global minimum at x = p if f p f x for all x in the domain. Any function that is continuous on a closed, bounded interval [a, b] will have both a global max and a global min on that interval. To find them a.) Find all the CPs in [a, b] b.) Find f(a) , f(b) and f(CP) for all CPs in [a, b] c.) The largest of these function values will be the global max and the smallest will be the global min. It’s important to remember that the global max/min might be at one of the endpoints or it might be at one of the CPs but it will exist. x5 16 x3 Example: Find the global max and min of the function f x on the 5 3 interval 7,5 . We already know there is a local max at x = -4 and a local min at x = 4. We want to find the function values at thest CPs and also at the endpoints: f 7 1532 f 4 136.53 f 4 136.53 f 5 41.67 We can see that on the interval 7,5 the global max occurs at x = -4 and the global min at x = -7.