Optimization I

Optimization I: Absolute Extrema

Definitions

  • Absolute Maximum: If ( f(c) \geq f(x) ) for all ( x ) in the domain of ( f ), then ( f(c) ) is called the absolute maximum of ( f ).

  • Absolute Minimum: If ( f(c) \leq f(x) ) for all ( x ) in the domain of ( f ), then ( f(c) ) is called the absolute minimum of ( f ).

Theorem 3

  • If a function ( f ) is continuous on a closed interval ( [a, b] ), then:

    • The absolute maximum and absolute minimum values exist on that interval.

General Fact

  • For a continuous function defined on a closed interval, the absolute maximum and minimum values are attained at:

    1. The critical numbers of the function.

    2. The endpoints of the interval.

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

  1. Evaluate ( f ) at:

    • (1) Critical numbers in the interval ( [a, b] ).

    • (2) The endpoints ( a ) and ( b ).

  2. The absolute maximum value corresponds to the maximum ( f ) value found.

  3. The absolute minimum value corresponds to the minimum ( f ) value found.

Examples

Example 1

  • Task: Find the absolute maximum and the absolute minimum value of a specific function (not provided in the transcript).

Results

  • Absolute Maximum Value: [Value not provided]

  • Absolute Minimum Value: [Value not provided]

Example 2a

  • Function: ( f(x) = x + 16x )

  • Domains:

    • (a) The domain ([1, 8])

    • (b) The domain ([1, 2])

  • Outcome: If the domain is not a closed interval, the absolute maximum and minimum values may not exist.

Example 2b

  • Function: ( f(x) = x + 16x )

  • Domain: ((0, \, \infty))

  • Task: Find the absolute extrema for ( f(x) ) given this domain (type to be covered in section 4.5).

Supplemental Examples

Example 5

  • Scenario: A stone is thrown straight up from the roof of an 80-ft building.

  • Height Function: ( h(t) = -16t^2 + 64t + 80 )

  • Question: What is the maximum height the stone reaches?

Example 6

  • Context: A manufacturer of tennis rackets.

  • Cost Function: ( C(x) = 400 + 4x + 0.0001x^2 )

  • Price Relation: Each racket sells at a price ( p ) given by the demand equation ( p = 10 - 0.0004x ).

  • Objective: Find the daily production level ( x ) that maximizes profit for the manufacturer.

Conclusion

  • The process of finding absolute extrema involves evaluating the function at critical points and boundaries within a closed interval. Continuous functions on such intervals guarantee the existence of absolute extrema, which can be pivotal for optimization problems across various applications.