Maxima and Minima: Fresh Take

  • Define and Identify the highest and lowest points of a function on a graph, both overall and within specific sections
  • Locate points on a function within a specific range where the slope is zero or undefined (critical points)
  • Explain how to use critical points to find the highest or lowest values of a function within a limited range

Extrema and Critical Points

Absolute Extrema

The Main Idea 

  • Absolute Extrema:
    • Absolute Maximum:
      • [latex]f(c) \ge f(x)[/latex] for all [latex]x[/latex] in the interval
    • Absolute Minimum:
      • [latex]f(c) \le f(x)[/latex] for all [latex]x[/latex] in the interval
    • Can be positive, negative, or zero
  • Extreme Value Theorem:
    • For a continuous function [latex]f[/latex] on a closed interval [latex][a,b][/latex]:
      • There exists a point where [latex]f[/latex] has an absolute maximum
      • There exists a point where [latex]f[/latex] has an absolute minimum
  • Conditions for Extrema:
    • Continuity is crucial for the existence of extrema
    • Closed, bounded intervals guarantee extrema for continuous functions
  • Types of Domains:
    • Infinite intervals may lack extrema
    • Open intervals may lack extrema even for continuous functions
    • Discontinuities can prevent extrema from existing

Analyze the function [latex]f(x) = x^3 - 3x[/latex] on the interval [latex][-2, 2][/latex] for absolute extrema.

Local Extrema and Critical Points

The Main Idea 

  • Local Extrema:
    • Local Maximum:
      • [latex]f(c) \ge f(x)[/latex] for all [latex]x[/latex] in some open interval containing [latex]c[/latex]
    • Local Minimum:
      • [latex]f(c) \le f(x)[/latex] for all [latex]x[/latex] in some open interval containing [latex]c[/latex]
  • Critical Points:
    • Interior point [latex]c[/latex] where [latex]f'(c) = 0[/latex] or [latex]f'(c)[/latex] is undefined
    • Candidates for local extrema
  • Fermat’s Theorem:
    • If [latex]f[/latex] has a local extremum at [latex]c[/latex] and [latex]f[/latex] is differentiable at [latex]c[/latex], then [latex]f'(c) = 0[/latex]
  • Key Relationships:
    • Local extrema occur only at critical points
    • Not all critical points yield local extrema
    • Endpoint extrema are not considered local extrema
  • Types of Critical Points:
    • Horizontal tangent line: [latex]f'(c) = 0[/latex]
    • Vertical tangent line or corner point: [latex]f'(c)[/latex] undefined

Find all critical points for [latex]f(x)=x^3-\frac{1}{2}x^2-2x+1[/latex].

Find the critical points of [latex]f(x) = x^{2/3}(x-2)[/latex] and determine if they correspond to local extrema.

Locating Absolute Extrema

The Main Idea 

  • Extreme Value Theorem:
    • A continuous function on a closed, bounded interval has both an absolute maximum and minimum
  • Location of Absolute Extrema:
    • Occur at endpoints of the interval or at critical points within the interval
  • Critical Points:
    • Points where [latex]f'(x) = 0[/latex] or [latex]f'(x)[/latex] is undefined
  • Procedure for Finding Absolute Extrema:
    • Evaluate function at endpoints 
    • Find and evaluate function at critical points within the interval 
    • Compare all values to determine absolute maximum and minimum
  • Interior vs. Endpoint Extrema:
    • Absolute extrema at interior points are also local extrema
    • Endpoint extrema are not considered local extrema

Find the absolute maximum and absolute minimum of [latex]f(x)=x^2-4x+3[/latex] over the interval [latex][1,4][/latex].

Find the absolute extrema of [latex]f(x) = x^3 - 6x^2 + 9x[/latex] on the interval [latex][0,4][/latex].