Function f(x) is positive, increasing and concave up on the closed interval [a, b]. the interval [a, b] is partitioned into 4 equal intervals and these are used to compute the upper sum, lower sum, and trapezoidal rule approximations for the value of integral b a f(x) dx. which one of the following statements is true?
lower sum < trapezoidal rule value < upper sum
lower sum < upper sum < trapezoidal rule value
trapezoidal rule < lower sum < upper sum
cannot be determined without the x-values for the partitions

The left sum would be f0+f1+f2+f3

The right sum would be f1+f2+f3+f4

The trapezoidal rule value is:

(f0+f1)/2 + (f1+f2)/2+(f2+f3)/2 +(f3+f4)/2

This would put the trapezoidal rule in the middle , which makes the

Lower sum < Trapezoidal rule Value < Upper sum

Step-by-step explanation:

Given function f(x) is positive, increasing and concave up on the closed interval [a, b],

it means f(x1) < f(x2) if x1 < x2

So Lower sum < Upper sum

As Trapezoidal is average of f(x1) and f(x2) = [f(x1) + f(x2)] / 2

it is average of the Lower and Upper sum.

The answer is Lower sum < Trapezoidal rule Value < Upper sum

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step-by-step explanation: