Section 4.1 Maximum and Minimum Values Applications of Differentiation

 Maxima and Minima  Applications of Maxima and Minima

Absolute Extrema Absolute Minimum Let f be a function defined on a domain D Absolute Maximum

The number f (c) is called the absolute maximum value of f in D A function f has an absolute (global) maximum at x = c if f (x)  f (c) for all x in the domain D of f. Absolute Maximum Absolute Extrema

Absolute Minimum Absolute Extrema A function f has an absolute (global) minimum at x = c if f (c)  f (x) for all x in the domain D of f. The number f (c) is called the absolute minimum value of f in D

Generic Example

Relative Extrema A function f has a relative (local) maximum at x  c if there exists an open interval (r, s) containing c such that f (x)  f (c) for all r  x  s. Relative Maxima

Relative Extrema A function f has a relative (local) minimum at x  c if there exists an open interval (r, s) containing c such that f (c)  f (x) for all r  x  s. Relative Minima

Fermat’s Theorem If a function f has a local maximum or minimum at c, and if exists, then Proof: Assume f has a maximum

The Absolute Value of x.

Generic Example The corresponding values of x are called Critical Points of f

Critical Points of f A critical number of a function f is a number c in the domain of f such that (stationary point) (singular point)

Candidates for Relative Extrema 1.Stationary points: any x such that x is in the domain of f and f ' (x)  0. 2.Singular points: any x such that x is in the domain of f and f ' (x)  undefined 3.Remark: notice that not every critical number correspond to a local maximum or local minimum. We use “local extrema” to refer to either a max or a min.

Fermat’s Theorem If a function f has a local maximum or minimum at c, then c is a critical number of f Notice that the theorem does not say that at every critical number the function has a local maximum or local minimum

Generic Example Two critical points of f that do not correspond to local extrema

Example Find all the critical numbers of Stationary points: Singular points:

Graph of Local max.Local min.

Extreme Value Theorem If a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and absolute minimum on [a, b]. Each extremum occurs at a critical number or at an endpoint. a b Attains max. and min. Attains min. but not max. No min. and no max. Open IntervalNot continuous

Finding absolute extrema on [a, b] 1.Find all critical numbers for f (x) in (a, b). 2.Evaluate f (x) for all critical numbers in (a, b). 3.Evaluate f (x) for the endpoints a and b of the interval [a, b]. 4.The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a, b], and the smallest value found is the absolute minimum for f on [a, b].

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max.Absolute Min. Evaluate Absolute Max.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Min. Absolute Max.

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Min. Absolute Max. Evaluate

Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Min. Absolute Max.