11th Lecture

Relative Maxima and Minima

  • Relative maxima are the high points on a graph compared to nearby points (e.g., points (c), (e), and (b) in the example graph).
  • Relative minima are the low points on a graph compared to nearby points (e.g., points (d), (k), and (a) in the example graph).
  • Collectively, these highs and lows are referred to as relative extrema.

Definitions

  • (f) has a relative maximum at (c) if there exists an interval ((r, s)) around (c) such that (f(c) \geq f(x)) for all (x) in ((r, s)).
  • (f) has a relative minimum at (c) if there exists an interval ((r, s)) around (c) such that (f(c) \leq f(x)) for all (x) in ((r, s)).

Absolute Maxima and Minima

  • Absolute maximum: The highest point on the entire graph of a function.
  • Absolute minimum: The lowest point on the entire graph of a function.

Stationary and Singular Points

  • Stationary point: A point (c) where (f'(c) = 0) (horizontal tangent line).
  • Singular point: A point (x) where (f'(x)) does not exist (e.g., a cusp).
  • Endpoints: The x-coordinates of the endpoints of the domain.

Procedure for Finding Extrema

  1. Find stationary points by solving (f'(x) = 0).
  2. Identify singular points where (f'(x)) does not exist.
  3. Consider the endpoints of the interval.
  4. Evaluate (f(x)) at all stationary points, singular points, and endpoints.
  5. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

Example: Finding Stationary Points

  • Given (f(x) = x^3 - 12x), the derivative is (f'(x) = 3x^2 - 12).
  • Factoring gives (f'(x) = 3(x - 2)(x + 2)).
  • Setting (f'(x) = 0) yields stationary points at (x = 2) and (x = -2).

Example: Finding Singular Points

  • Given (f(x) = 3(x - 1)^{1/3}), the derivative is (f'(x) = (x - 1)^{-2/3}).
  • (f'(x)) does not exist when (x = 1), so there is a singular point at (x = 1).

Example: Finding Extrema on an Interval

  • Given (f(x) = x^2 - 2x) on the interval ([0, 4]), the derivative is (f'(x) = 2x - 2).
  • Setting (f'(x) = 0) gives a stationary point at (x = 1).
  • No singular points exist because (f'(x)) is defined everywhere.
  • Endpoints are (x = 0) and (x = 4).
  • Evaluate (f(0) = 0), (f(1) = -1), and (f(4) = 8).
  • Relative minimum at (x=1). Absolute maximum at (x=4).

Key Points

  • Stationary points, singular points, and endpoints are candidates for relative extrema.
  • Graphing technology combined with calculus provides a precise way to find extrema.