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Relative Maxima and Minima Eric Hoffman Calculus PLHS Nov. 2007 Key Topics • Critical Numbers: the x-values at which the f ‘(x)=0 or f ‘(x) fails to exist • Note: The critical numbers are the points where the graph will switch from increasing to decreasing or vice versa • Find the critical numbers for the following functions: f(x) = 3x2 – 6x + 3 x=1 f(x) = x3/2 – 3x + 7 x=4 Key Topics • Relative maximum: the highest value for f(x) at that particular “peak” in the graph • Relative minimum: the lowest value for f(x) at that particular “valley” in the graph Relative maximum Relative maximum Relative minimum Relative minimum Relative minimum Key Topics • How to determine whether it is a relative maximum or a relative minimum at a focal point: Step 1: Find the focal points of the graph to determine the intervals on which f(x) is increasing or decreasing Step 2: Choose an x-value in each interval to determine whether the function is increasing or decreasing within that interval Step 3: If f(x) switches from increasing to decreasing at a focal point, there is a relative maximum at that focal point If f(x) switches from decreasing to increasing at a focal point, there is a relative minimum at that focal point Key Topics + to - means maximum means minimum •- to It + might help to make a number line displaying your findings - - - | +++++++ | - - - - | +++++++ | - - - - - - - | +++ • Another helpful method might be to make a 3 – 3x2 – 12x + 1 table of your findings f(x) This tells us that=in2x this interval the Interval (-∞,-1) (-1,2) (2,∞) function is increasing us that in this interval the TestThis # tells ‘(t) Sign function isf decreasing us that in this interval the t =This -2 tells 24 function is increasing t=0 t=3 -12 24 of f ‘(t) + + Key Topics • Homework: pg. 186 1 – 22 all