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)