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Extreme Value Theorem
EXTREME VALUE THEOREM Let f he continuous on its domain, which is a closed interval [a, b]. Then f has a maximum at some point in [a, b], and a minimum at some point in [a, b]. Discussion We have seen several examples of functions that do not have maxima on an open interval, such as f(x) = l/x on (0, ∞), or g(x) = 2x on (0, 1). The Extreme Value Theorem says that on a closed interval a continuous function always has a maximum. SKETCH OF PROOF Form an infinite partition of [a, b]*, a, a + δ, a + 2δ,..., a + Hδ = b. By the Transfer Principle, there is a partition point a + Kδ at which f(a + Kδ) has the largest value. Let c be the standard part of a + Kδ (see Figure 3.8.10). Any point u of [a, b]* lies in a subinterval, say a + Lδ ≤ u < a + (L + 1)δ. We have f(a + Kδ) ≥ f(a + Lδ), and taking standard parts, f(c) ≥ (u). This shows that f has a maximum at c.
Figure 3.8.10 Proof of the Extreme Value Theorem
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Last Update: 2006-11-05