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Department

Mathematics

Course Code

MATH-M 119

Professor

Tracy Whelan

Lecture

25

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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