Probem+2+James,+Kevin

Kevin Wu: M=xy product is a minimum x-y=100 difference is 100
 * 1) 2. Find two numbers whose difference is 100 and whose product is a minimum.

Substitute: for x=100+y M=xy M=y(100+y) M=100y+y^2

Find critical points by taking the derivative of this function and setting it to zero M'=100+2y 0=100+2y -100=2y y=-50

Substitute this into x-y=100 to find the other number: y=-50 x-y=100 x-(-50)=100 x=50

The two numbers are: 50 and -50.

Good job! Ms. Sweeney Actually, I am changing my mind here -- all you have shown is that we have a critical value at x = 50. How do we know that this x-value yields an ABSOLUTE minimum product? What about the endpoints? Lim y = infinity ± 100 x→infinity

Therefore xy = large positive #

Lim y = - infinity ± 100 x→ - infinity

Therefore xy = large positive # Since the objective function is M = y² + 100y, we know the graph is a parabola opening upwards so the graph will not drop back down. Also, as x approaches ± infinty, y approaches infinity. Therefore there is a absolute minimum at x = 50 and the two numbers are 50 and -50.

ALSO in case that wasn't enough, the second derivative of the function M = y² + 100y is just 2. Since the second derivative is always positive, the graph of the function will always be concave up.