Function f(x) is positive, decreasing 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 left sum, right sum, and trapezoidal rule approximations for the value of integral from a to b f(x)dx. which one of the following statements is true?
a) left sum < trapezoidal rule value < right sumb) left sum < right sum < trapezoidal rule valuec) right sum

Right sum < trapezoidal rule value < Left sum

Non of the completed options are valid. Option c is incomplete but it starts correctly. If there are no more options, the answer is (c).

Step-by-step explanation:

Lets call a = x0, x1, x2, x3 and x4 = b the endpoints of the intervals in increasing order. Lets also call L the length of each subinterval. Note that b-a = 4L. Lets denote yi = f(xi) for i in {0,1,2,3,4}.

Lets compute each sum:

Left sum = x0*L + x1*L + x2*L + x3*L = (y0+y1+y2+y3)L

Similarly

Right sum = (y1+y2+y3+y4)L

and trapezoidal rule value = L((y0)/2 + y1+ y2 + y3+ (y4)/2)

Since f is decreasing, we have that y0 > y1 > y2 > y3 > y4. Therefore, y0 > y4 and, as a result, Left sum > Right sum, because the Right sum has y4 instead of y0 multiplying by L.

On the other hand, multiplying L we have (y0)/2 + y1+ y2 + y3+ (y4)/2 on the trapezoidal rule value, thus it has half of y0 and half of y4, and as a consecuence, the trapezoidal sum is between the left sum and the right sum.

The correct answer is

Right sum < trapezoidal rule value < Left sum

answer is c, i took the test

Step-by-step explanation:

answer: 1) twenty-seven times y

              2) twenty-seven multiplied by y

c.

step-by-step explanation:

because first you have to add the 4x, 9x, and the 7x together and you get 20x and then you add the last numbers up and you get 5and then you put the subtraction sign in there and that's how you get the answer.