4.1 Maximum and Minimum Values

Maximum Values

Local Maximum

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Absolute Maximum

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c2

|c1

I

Minimum Values

Local Minimum

•

Absolute Minimum

•I

c2

||c1 I

Collectively, maximum and minimum values are called extreme values.

DefinitionsA function f has an absolute (global) maximum at c if f (c) ≥ f (x) for all x in the domain.

A function f has an absolute (global) minimum at c if f (c) ≤ f (x) for all x in the domain.

The maximum and minimum values are called extreme values.

A function f has a local (relative) maximum at c if f (c) ≥ f (x) where x is in a small open interval about c.

A function f has a local (relative) minimum at c if f (c) ≤ f (x) where x is in a small open interval about c.

The Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f (c) and an absolute minimum f (d) at some numbers c and d in [a, b].

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Fermat’s Theorem

If f has a local maximum or minimum at c, and

if f '(c) exists, then f '(c) = 0.

Question If a (local) maximum or minimum occur at c, then what is the value of f '(c)?

Critical Number or Value

A critical number of a function f is a number c in the domain of f such that either f '(c) = 0 or f '(c) does not exist ( f (x) is not differentiable)

Fact

An absolute extremum occurs at two places: Critical points End points

Finding Absolute Extrema on a Closed Interval [a,b]

1. Find the critical numbers of f on (a, b).

2. Compute the value of f at each of • the critical numbers on (a,b) • the endpoints a and b.

3. The largest of these values is the absolute maximum. The smallest is the absolute minimum.

Examples

Locate the absolute extrema of the function on the closed interval.

f(x) = x3 – 12x on [-3,4]

g(x) = 4x / (x2+1) on [0,3]

h(t) = 2 sec(t) - tan(t) on [0,π/4]

4 3 2( ) 4 4 1 [0,2]f t t t t on 8 2

3 3( ) 16 3 [ 3,1]h x x x on