Lecture Notes on Finding Local Maximum and Minimum Values of Functions

Key Concepts

• Local Maximum/Minimum: Extreme values where the first derivative equals zero.
• Critical Numbers: Values of x where the derivative is zero, indicating potential maxima or minima.
• Sign Charts: Used to determine if a critical number is a maximum or minimum.

Methodology

1. Find First Derivative: Derive the function to find f'(x).
2. Set Derivative to Zero: Solve f'(x) = 0 to find critical numbers.
3. Use Sign Chart:
   • Test intervals around critical numbers to determine sign changes.
   • A change from negative to positive indicates a local minimum.
   • A change from positive to negative indicates a local maximum.

Example Problems

Example 1: ( f(x) = x^2 - 4x )

• Find First Derivative: f'(x) = 2x - 4.
• Solve for Critical Number: 2x - 4 = 0 ⇒ x = 2.
• Determine Sign Changes:
  • Test values around x = 2.
  • Changes from negative to positive, indicating a local minimum.
• Evaluate Function at x=2: ( f(2) = -4 ).
• Result: Local minimum at (2, -4).

Example 2: ( f(x) = 2x^3 + 3x^2 - 12x )

• First Derivative: f'(x) = 6x^2 + 6x - 12.
• Factor and Solve: 6(x+2)(x-1) = 0 ⇒ x = -2, 1.
• Sign Chart Analysis:
  • Positive to negative at x = -2: Local maximum.
  • Negative to positive at x = 1: Local minimum.
• Evaluate Function Values:
  • Local max at (-2, 20).
  • Local min at (1, -7).

Example 3: ( f(x) = 3x^4 - 16x^3 + 24x^2 )

• First Derivative: f'(x) = 12x(x^2 - 4x + 4).
• Factor and Solve: 12x(x-2)^2 = 0 ⇒ x = 0, 2.
• Sign Chart Analysis:
  • Minimum at x = 0 (decreasing to increasing).
  • x = 2 shows no sign change (neither max nor min).
• Result: Only minimum at x = 0.

Important Notes

• Multiplicity: Even multiplicity in factors implies no sign change.
• Always confirm sign chart results by testing values and understanding function behavior.
• Ordered pairs for extrema are often necessary depending on problem requirements.