Q: Which of the following are true, if 0 < a < 1 and 0 < b < 1?
UNANSWERED.
The correct answer is B.
A:
The answer is B.
√ a * √ b = √ (a * b) , so statement I is always false.
Let's pick a number for a and b, say a = 1/4 and say b = 1/9. Then let's plug it into statement II. √ a + √ b > √ (a + b) √ 1/4 + √ 1/9 > √ (1/4 + 1/9) 1/2 + 1/3 > √ (13 / 36) 5/6 > √ (13/36) Use the calculator to get .8333... > .6009... This shows that statement II is true at least sometimes.
For statement III, let's pick some numbers that are easy to calculate a = 1/4 and b = 1/2. √ a + √ 2b < √ (a + 2 b) √ 1/4 + √ 1 < √ (1/4 + 2 * 1/2) 1/2 + 1 < √ (5/4) Use your calculator to find that the approximate value of √ 5/4 is 1.11. 1.5 < 1.11 This shows that statement III is false at least once (when we set a = 1/4 and b = 1/2), so it is a false statement.
Statement II is the only one that might be true. We know it works in at least one case (remember, where a = 1/4 and b = 1/9). It still might be false for another case, but we haven't proven it either way. Looking at the possible answers, we eliminate choices A), C), D), and E) because we know I and III are false. B) II only must be correct.
