(3) Let R be an n-dimensional rectangular box and let f(x1,.. , xn) be continuou
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Question
(3) Let R be an n-dimensional rectangular box and let f(x1,.. , xn) be continuous on R. Assume in addition that f(x1, ... , xn) > = 0 for all (x1, .... , xn) element of R. (a) Show that Integrate ... integrate R f(x1,...., xn)d(x1, ...., xn) > = 0. (b) Assume that Integrate ... integrate R f(x1,...., xn)d(x1, ...., xn) = 0. Show that f(x1,...., xn) for all (x1, ...., xn) element of R. (c) Is the conclusion in part b still valid if f is only integrable, but not necessarily continuous, on R? Justify your answer.Explanation / Answer
a)
as f(xn) is continuous on R then double integral f(xn)d(xn) will be greater or equal to 0 as f(xn)>=0 for all xn E R
if the function is = 0 then its integral will be zero as wel
c)
the conclusion in part b will not hold true if the function is just integrable and not everywhere continuous on R.
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