Question: Use the fact that a continuous function on a closed bounded domain D attains both a minimum value m and a maximum value M, together

Use the fact that a continuous function on a closed bounded domain D attains both a minimum value m and a maximum value M, together with Theorem 3, to prove that the average value lies between m and M. Then use the IVT in Exercise 68 to prove the Mean Value Theorem for Double Integrals.

THEOREM 3 Let f(x, y) and g(x, y) be integrable functions on D. (a) If f(x,y) g(x, y) for all (x, y) e D,

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