Limits & Continuity · Continuity & Theorems
Extreme Value Theorem
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.
- 0 variables
- 2 worked examples
- 18 in Limits & Continuity
Worked examples
Find the absolute extrema of f(x) = x² on [-1, 3].
- f is a polynomial (continuous). Apply EVT: extrema exist.
- f'(x) = 2x = 0 → x = 0 (critical point in [-1,3])
- Evaluate: f(-1) = 1, f(0) = 0, f(3) = 9
Answer: Absolute min = 0 at x = 0; absolute max = 9 at x = 3.
1 more worked examplePremium
Find the absolute extrema of f(x) = sin(x) on [0, 2π].
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Practice
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