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Relative Maxima and Minima Eric Hoffman Calculus PLHS Nov. 2007

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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) = 3x 2 – 6x + 3f(x) = x 3/2 – 3x + 7 x = 1 x = 4

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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 minimum

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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

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Key Topics It might help to make a number line displaying your findings - - - | +++++++ | - - - - | +++++++ | - - - - - - - | +++ - to + means minimum + to - means maximum

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Another helpful method might be to make a table of your findings f(x) = 2x 3 – 3x 2 – 12x + 1 IntervalTest #f (t)Sign of f (t) (-,-1)t = -224+ (-1,2)t = 0-12- (2,)t = 324+ This tells us that in this interval the function is increasing This tells us that in this interval the function is decreasing This tells us that in this interval the function is increasing

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Key Topics Homework: pg. 186 1 – 22 all

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Increasing and Decreasing Functions

Increasing and Decreasing Functions

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