Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics II / 6 / d / i

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- Solution i

Solution

The extreme value theorem gives finite real numbers

α = min_{x \in [a, b]} f (x), β = max_{x \in [a, b]} f (x), α \leq β .

Since these values are attained, the set of strict upper bounds in the definition of

M

is precisely

(β, \infty)

, and the set of strict lower bounds in the definition of

m

is precisely

(- \infty, α)

. They are nonempty and bounded on the required sides. By the definitions of infimum and supremum,

M = in f (β, \infty) = β, m = sup (- \infty, α) = α, m \leq M .

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