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Limits & Continuity · Continuity & Theorems

Extreme Value Theorem

f continuous on [a,b]⇒f attains an absolute max and min on [a,b]f \text{ continuous on } [a,b] \Rightarrow f \text{ attains an absolute max and min on } [a,b]

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on that interval.

Conditions. f must be continuous and the interval must be closed and bounded.

Worked examples

Find the absolute extrema of f(x) = x² on [-1, 3].
  1. f is a polynomial (continuous). Apply EVT: extrema exist.
  2. f'(x) = 2x = 0 → x = 0 (critical point in [-1,3])
  3. Evaluate: f(-1) = 1, f(0) = 0, f(3) = 9

Answer: Absolute min = 0 at x = 0; absolute max = 9 at x = 3.

Find the absolute extrema of f(x) = sin(x) on [0, 2π].
  1. f is continuous. Critical points where cos(x) = 0: x = π/2, 3π/2
  2. f(0) = 0, f(π/2) = 1, f(3π/2) = -1, f(2π) = 0

Answer: Absolute max = 1 at x = π/2; absolute min = -1 at x = 3π/2.

Related formulas

Intermediate Value Theorem

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