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Relative Maxima and Minima

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

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Relative Maxima and Minima

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  1. Relative Maxima and Minima Eric Hoffman Calculus PLHS Nov. 2007

  2. 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 f(x) = x3/2 – 3x + 7 x = 1 x = 4

  3. 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

  4. 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

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

  6. This tells us that in this interval the function is increasing This tells us that in this interval the function is decreasing • Another helpful method might be to make a table of your findings f(x) = 2x3 – 3x2 – 12x + 1 This tells us that in this interval the function is increasing

  7. Key Topics • Homework: pg. 186 1 – 22 all

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