Find two positive numbers whose sum is 190 and whose product is a maximum. evaluate the remaining asked questions.

(a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) First Second Number, x Number Product, P 50 190 - 50 50(190 - 50) = 7,000 50 190 - 60 60(190 - 60) = 7,800 70 190 - 70 70(190 - 70) = 80 190 - 80 80(190 - 80) = 90 190 - 90 90(190 - 90) = 100 190 - 100 100(190 - 100) = (b) Write the product P as a function of x. P (x ) = (c) Use calculus to find the critical number of the function in part (b). Then find the two numbers. (Enter your answers as a comma-separated list.) (d) Use a graphing utility to graph the function in part (b) and verify the solution from the graph. 8000 10000 9000 150 6000- 8000- 8950 100- 4000- 6000 8900 2000- 4000 - 2000 8950 - 20 40 60 80 100 120 140 160 180 200 O 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 -2000 20 40 60 80 100 120 140 160 180 200 The solution appears to be at x =