In-Depth Notes on Absolute Extrema and Local Extrema

Absolute Extrema

  • Definition:
    • Absolute Extrema (Points): Points where the function takes its largest or smallest values on a closed interval.
    • Strict Definition:
      • Absolute maximum at point (c, f(c)) if for all x: f(x) ≤ f(c).
      • Absolute minimum at point (c, f(c)) if for all x: f(x) ≥ f(c).

Finding Local Extrema

  • First Derivative Test:
    • Local Maximum Conditions: f'(x) = 0 and f''(x) < 0.
    • Local Minimum Conditions: f'(x) = 0 and f''(x) > 0.
Example (Finding Local Extrema):
  • Given function: f(x) = -3x²
    • First Derivative: f'(x) = -6x
    • Set f'(x) to zero:
      • -6x = 0
      • Solution: x = 0
    • Second Derivative: f''(x) = -6
      • Since f''(x) < 0, there is a local maximum at x = 0.
    • Value at x = 0: f(0) = 0
    • Therefore, max point is (0, 0).

Conditions for Absolute Extrema

  • Evaluate Function at Endpoints:
    • If endpoints are part of the interval, check function values at those points.
  • Compare Values:
    • Confirms if a point is absolute max/min by examining endpoints and local extrema.
Example 1 (Finding Absolute Extrema):
  • Given function: f(x) = x² + 2
    • Evaluating Extremes:
      • Absolute max: none (as f(x) reaches infinity)
      • Absolute min: (0, 2)
  • Analysis: All values approach infinity as x moves away from 0.
Example 2:
  • Given function: f(x) = 4 - x² (upside down parabola)
    • Finding Extrema:
      • f'(x) = -2x
      • Set f'(x) to zero: -2x = 0 → x = 0
      • Value at x = 0: f(0) = 4
    • Thus, (0, 4) is the absolute max.
    • No absolute min exists as the function tends to negative infinity.

Summary of Steps to Find Absolute Extrema:

  1. Compute the first derivative f'(x) and find critical points (set f'(x) to 0).
  2. Compute the second derivative f''(x) to test for local maxima or minima.
  3. Evaluate function values at endpoints of the interval.
  4. Compare all values (critical points & endpoints) to determine absolute extrema.
  5. Identify if maximums or minimums occur at endpoints as well as critical points.