Which of the following statements is true?
2000
Which of the following statements is true?
- A.
S > T
- B.
S = T
- C.
S < T and 2S > T
- D.
2S <= T
Show answer & explanation
Correct answer: A
Let f(x) = x log2(x). For x >= 2, f is increasing because f'(x) = log2(x) + 1/ln 2 > 0.
For each integer i = 3, 4, ..., 100 and every x in [i - 1, i], we have f(x) <= f(i). Therefore the integral from i - 1 to i of f(x) dx is less than f(i).
Adding these inequalities gives: T = integral from 2 to 100 of x log2(x) dx < sum from i = 3 to 100 of i log2(i) = S.
Hence S > T. Option A is correct.