Please show work.

Problem 1: Find two numbers whose difference is 100 and whose produce is a minimum. Problem 2: The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? Problem 3: Find the dimension of a rectangle with area 1000 m" whose perimeter is as small as possible. Problem 4: The rate (in mg carbon/ mo/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the infection 100/ P Z + + 4