Among all pairs of numbers whose difference is 58 find a pai

Among all pairs of numbers whose difference is 58, find a pair whose product is as small as possible. 87 and 29 29 and 29 -87 and -29 -29 and 29

Solution

Let the smaller of the 2 numbers be x. Then we have to compute the mimimum value of x (x+58)= y (say). We know that if y is minimum then dy/dx = 0 and d²y/dx² is positive. Here, y = x(x +58) = x² +58x so that dy/dx = 2x +58. Hence if dy/dx = 0, then 2x = -58 so that x = -29. Also, d²y/dx² = 2 which is positive. Hence, for x(x + 58) to be minimum, we must have x = -29. Further, if x = -29, then x + 58 = -29 + 58 = 29. Thus, option 4) is the correct answer.