Find two numbers whose difference is 102 and whose product is a minimum. Step 1 If two numbers have a difference of 102, and one of them is x + 102, then the other is $ Incorrect: Your answer is incorrect. x. Step 2 The product of two numbers x and x + 102 can be simplified to be x2 Correct: Your answer is correct. seenKey 2 + 102 Correct: Your answer is correct. seenKey 102 x. Step 3 If f(x) = x2 + 102x, then f '(x) = $ Correct: Your answer is correct. 2x+102. Step 4 To minimize the product f(x) = x2 + 102x, we must solve 0 = f '(x) = 2x + 102, which means x = -51 Correct: Your answer is correct. seenKey -51 . Step 5 Since f ''(x) = 2 , there must be an absolute minimum at x = −51. Thus, the two numbers are as follows. (smaller number) (larger number)