Show that if f: [0, ) R is continuous, satisfies f(x) > 0 for all x...

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Show that if f: [0, ∞)→ R is continuous, satisfies f(x) > 0 for all x ≥ 0 and lim f(x) = 0, then f attains its maximum. 14x Hint: We want to be able to restrict our attention to a compact subset of [0, ∞) and apply the extreme value theorem there. Show that if f: [0, ∞)→ R is continuous, satisfies f(x) > 0 for all x ≥ 0 and lim f(x) = 0, then f attains its maximum. 14x Hint: We want to be able to restrict our attention to a compact subset of [0, ∞) and apply the extreme value theorem there.