 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-ii/6/d/i/solution

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)

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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