1983 AHSME Problems/Problem 21
Find the smallest positive number from the numbers below.
Notice that , , and similarly, , so these numbers can be excluded immediately as they are negative and we seek the smallest positive number. The remaining choices are and . Observe that , and , so the smallest positive number in the list is .
After reaching the two possibilities and as in the first solution, note that these can be written as for and respectively. Now, (as the numerator is always negative and the denominator is always positive for ). Therefore, with its derivative being negative, is a strictly decreasing function over its domain, so the smaller value will be the one with the larger value of , namely . Therefore the answer is as before.