Mathematics

Mathematics

Problem

Let ff be a positive, decreasing function on [1,5][1,5]. A sum is formed by multiplying the width of each subinterval in a partition 1=x0<x1<x2<x3=51=x_0<x_1<x_2<x_3=5 by the value of ff at the left endpoint of that subinterval and then adding the results. Which statement best describes this sum relative to ∫15f(x) dx\displaystyle \int_{1}^{5} f(x)\,dx? a) It is an underestimate because ff is decreasing on [1,5][1,5]. b) It could be an overestimate or an underestimate depending on the concavity of ff on [1,5][1,5]. c) It is an overestimate because ff is decreasing on [1,5][1,5]. d) Whether it is an overestimate or an underestimate cannot be determined without a uniform partition.

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