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Week 12a - QMA 2006S2 (Maxima Minima &amp; Integration)

Week 12a - QMA 2006S2 (Maxima Minima &amp; Integration)...

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Jessica Chung & Derek Hui Week 12 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] 1/4 M AXIMA AND M INIMA OF F UNCTIONS Most of you would have covered this in high school, if you haven’t let us know!! There are two distinct types of maxima and minima (just some funky terms to remember), a global/absolute maximum/minimum and a local/relative maximum/minimum: Given ) ( x f y = and a x = is a point in the domain of ) ( x f Global or absolute maximum occurs if ) ( ) ( x f a f for all x in the domain of ) ( x f Global or absolute minimum occurs if ) ( ) ( x f a f for all x in the domain of ) ( x f Local or relative maximum occurs if ) ( ) ( x f a f for all x in some interval of ) ( x f Local or relative minimum occurs if ) ( ) ( x f a f for all x in some interval of ) ( x f However this is best illustrated visually: NOTE: An extremum is a term used for both maximum and minimum points (comes from “extremity”). Hence, a global or absolute extremum is always a local or relative extremum as well. F INDING E XTREMA First Derivative Test: Find Stationary Points: o Solve 0 ) ( ' = x f or 0 = dx dy to obtain stationary points Check if stationary points are maxima or minima: o “Pidgeon hole” test o See if ) ( x f is increasing [ ] 0 ) ( ' > x f or decreasing [ ] 0 ) ( ' < x f on either side of the stationary point o If ) ( ' x f goes from positive to negative as x increases it is a local maximum o If ) ( ' x f goes from negative to positive as x increases it is a local minimum o If ) ( ' x f doesn’t change – may be a point of inflection (sometimes spelt “inflexion”) Second Derivative Test: Find Stationary Points: o Solve 0 ) ( ' = x f or 0 = dx dy to obtain stationary points ( turning points )

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Jessica Chung & Derek Hui Week 12 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] 2/4 Calculate the second derivative: ) ( ' ' x f or 2 2 dx y d at each stationary point: o If 0 ) ( ' ' < x f then it is a local maximum o If 0 ) ( ' ' > x f then it is a local minimum o If 0 ) ( ' ' = x f then we must check if it is a point of inflection Points of Inflection:
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