Lecture Notes: Maximum and Minimum Values

Key Concepts

Absolute Maximum and Minimum

• Absolute Maximum: A function ( f ) has an absolute maximum at ( C ) if ( f(C) \geq f(X) ) for all ( X ) in the domain of ( f ).
• Absolute Minimum: A function has an absolute minimum at ( K ) if ( f(K) \leq f(X) ) for all ( X ) in the domain.

Local (Relative) Maximum and Minimum

• Local Maximum: ( f ) has a local maximum at ( B ) if ( f(B) \geq f(X) ) when ( X ) is near ( B ) (or in an open interval containing ( B )).
• Local Minimum: ( f ) has a local minimum at ( M ) if ( f(M) \leq f(X) ) when ( X ) is near ( M ).

Graphical Interpretation

• Local Max/Min Identification: Points on a graph where the behavior of the function changes direction indicating peaks or troughs.
• Endpoints: Not considered for local max/min values due to lack of neighbors on one side but can be absolute max/min.

Example Analysis

• Graph Interpretation:
  • Identify maxima and minima by examining the open intervals and endpoints.
  • Ex: ( f(B) ) is a local max, ( f(F) ) is both a local and absolute min, and ( f(G) ) as an endpoint is an absolute max.

Sketch and Analyze Functions

• Process: Sketch a graph - a parabola in this case - and determine extrema by evaluating function values at given points.
• Example: Parabola shifted and domain restricted from [-2,2].
  • Open circle indicates no absolute max.
  • Identify absolute min at ( f(-1) = 1 ) and local min also at ( f(-1) = 1 ).

Theorems and Definitions

Extreme Value Theorem

• Conditions: Continuous function on a closed interval ([A, B]).
• Guarantee: Function will have an absolute max and min on this interval.

Fermat's Theorem

• Statement: If ( f ) has a local max/min at ( C ) and ( f' ) exists, then ( f'(C) = 0 ).
• Implication: Slope of tangent line at local extrema is horizontal (0 slope).

Critical Numbers

• Definition: ( C ) is a critical number if ( f'(C) = 0 ) or ( f'(C) ) does not exist, provided ( C ) is in the function's domain.
• Relation to Extrema: Local extrema occur at critical numbers but not all critical numbers indicate local extrema.

Example: Finding Critical Numbers

• Function: ( H(P) = \frac{P-1}{P^2 + 4} ).
• Steps:
  • Calculate derivative ( H'(P) ) using quotient rule.
  • Solve ( H'(P) = 0 ) to find zeros.
  • Check where ( H'(P) ) does not exist.
• Solution: Critical numbers found at ( 1 \pm \sqrt{5} ) since no points where denominator equals zero.

Conclusion: Understanding derivatives and critical numbers helps locate maximum and minimum values, essential for analyzing function behavior.