Problem+1+Sam+B,+Ben+Z.

very good!

Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum.

a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem.

Given Table: I didn't really think it was necessary to continue this. If you do this table based off of integers, you're going to find that your greatest product is going to be when your two numbers are 11 and 12.
 * First number || Second number || Product ||
 * 1 || 22 || 22 ||
 * 2 || 21 || 42 ||
 * 3 || 20 || 60 ||

b) Use calculus to solve the problem and compare with your answer to part a.

x+y=23 xy= Max So you are trying to Maximize M = xy, where x + y = 23 y=23 - x x(23-x)= M 23x - x^2 = M M'= -2x+23 0= -2x+23 2x= 23 x= 23/2 or 11.5 So if we do not have to use integers, both numbers would be 11.5

To justify this, second derivative: m''= -2 by 2nd deriv, 11.5 is a (local) max

The limit as x approaches infinity is negative infinity. As x approaches negative infinity the limit is also negative infinity. Therefore you can eliminate the endpoints. or rather, the possibility that the absolute max occurs at the endpoints.