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 Calculus I - Condensed Class Notes and Problems - Tristan Whalen Page 1 Calculus I - Condensed Notes 4.3 Local Extreme Values A function has a local maximum at x = c if it switches from increasing to decreasing there. The local max value is F(c). A function has a local minimum at x = c if it switches from decreasing to increasing there. The local min value is Fle). Critical numbers are candidates for the locations of local extreme values. These are places where f'(x) = 0 or where f'(x) is undefined. There are two major steps to finding and classifying local extreme values: find the critical numbers, and then apply either the first derivative test or the second derivative test to each critical number. First Derivative Test: 1) Find where the function f is undefined. 2) Find f(x) and see where it is undefined. 3) Set f"(x) = 0 and solve for x. 4) Mark the numbers from steps 1,2, and 3 on the number line. This breaks it up into intervals, S) Check the sign of f'(x) on each interval in step 4 by using a convenient "test point" in the interval. Mark whether f'(x) is positive or negative on each interval. 6) Look at each number from steps 2 and 3. If "' (x) switches from positive to negative at x = c, then f(c) is a local maximum. If f' (x) switches from negative to positive at x = c, then f(c) is a local minimum. If f (c) is undefined, then it cannot be a local max or min. Second Derivative Test: When to use this instead of the first derivative test: If the function and its derivatives are defined everywhere and the second derivative is easy to find. critical numbers. If you only need to classify one critical number e for which /"(c) = 0 and, perchance, don't know the other 1) This works only on the critical numbers x = c where f'(c) = 0. 2) Find f"(x) and plug in c. 3) if f"(c) > 0, then f(c) is a local minimum. If f"(c) < 0, then f(c) is a local maximum. If f"(c) = 0 then you cannot draw any conclusion; use the first derivative test. Problems: Find and classify the local extreme values of the following functions: 1) f(x) = 2x3 - 5x2 - 4x + 2 2) f(x) = - x7 + 9x2 - 24x + 18 3) f(x) = |x2 - 1| 4) f(x) = x2 + 1 + 6 5) Suppose that x = 2 is a critical number for f and f'(x) = 2x3 - x2 + 3.x - 18. Classify f (2) as a local maximum, local minimum, or neither.