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Lecture 25

MATH-M 119 Lecture Notes - Lecture 25: Maxima And Minima, Microsoft Powerpoint
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Fall 2018

Department
Mathematics
Course Code
MATH-M 119
Professor
Tracy Whelan
Lecture
25

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MATH M119 - Lecture 25 Local Minima and Maxima
Local Minima: F has a local minimum at p if F(p) is less than or equal to the values of F for
points near p
Local Maxima: F has a local maximum at p if F(p) is greater than or equal to the values of F for
points near p
Critical Point: for any function F, a point p in the domain of F where   or is
undefined
Point (p, F(p)) is also a critical point
If p is a critical point of F, then F(p) is called a critical value of F
Geometrically, at a critical point p, a function F has either:
o A horizontal tangent: when  
o No tangent or vertical tangent (when  is undefined
If a function F which is continuous on interval has a local maximum or minimum at p,
then p is either a critical point of F or an endpoint of the interval
o Not every critical point is a local maximum or minimum; may have horizontal
tangent without maximum or minimum
Example (PowerPoint Slide 5):
This function has 2 critical points at (-2,4) and (0,0)
First Derivative Test for Local Maxima and Minima
Suppose p is a critical point of a continuous function F
F changes from decreasing to increasing at p <-> F has local minimum at p
o F’ changes sign from – to +
F changes from increasing to decreasing at p <-> F has a local maximum at p
o F’ changes sign from + to –

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Description
MATH M119 - Lecture 25 Local Minima and Maxima Local Minima: F has a local minimum at p if F(p) is less than or equal to the values of F for points near p Local Maxima: F has a local maximum at p if F(p) is greater than or equal to the values of F for points near p Critical Point: for any function F, a point p in the domain of F where = 0 or is undefined Point (p, F(p)) is also a critical point If p is a critical point of F, then F(p) is called a critical value of F Geometrically, at a critical point p, a function F has either: o A horizontal tangent: when = 0 o No tangent or vertical tangent (when () is undefined If a function F which is continuous on interval has a local maximum or minimum at p, then p is either a critical point of F or an endpoint of the interval o Not every critical point is a local maximum or minimum; may have horizontal tangent without maximum or minimum Example (PowerPoint Slide 5): This function has 2 critical points at (-2,4) and (0,0) First Derivative Test for Local Maxima and Minima Suppose p is a critical point of a continuous function F F changes from decreasing to increasing at p F has local minimum at p o F changes sign from to + F changes from increasing to decreasing at p F has a local maximum at p o F changes sign from + to
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