For positive integers x and y, x+y=21. What is the smallest possible value of xy?

a.10
b.20
c.38
d.54
e.110

The largest value is when x and y are almost equal: 10*11

The smallest value is when they are as different as possible: 1*20

20

To find the smallest possible value of xy, we need to find the values of x and y that minimize their product.

Since x+y=21, we can rewrite this equation as x = 21 - y.

Substituting this expression for x into the product xy, we get (21 - y) * y.

To find the smallest possible value of xy, we can find the minimum value of the quadratic equation (21 - y) * y.

To do this, we can find the vertex of the quadratic equation.

The vertex of the quadratic equation y = ax^2 + bx + c is given by the x-coordinate (-b/2a).

In this case, the quadratic equation is (21 - y) * y.

So, a = -1, b = 21, and c = 0.

Plugging these values into the formula for the x-coordinate of the vertex, we get:

x-coordinate of the vertex = -b / (2a) = -21 / (-2) = 10.5.

Since x and y are positive integers, the closest values for x and y that yield a value close to 10.5 are 10 and 11.

So, the product xy is equal to 10 * 11 = 110.

Therefore, the answer is e. 110.

To find the smallest possible value of xy, we need to consider the relationship between x and y. Since x and y are positive integers, their sum, x+y, is minimized when x is as small as possible and y is as large as possible. In this case, x would be 1 and y would be 20, because 1+20=21.

Now, we can calculate the value of xy. xy = 1*20 = 20.

Therefore, the smallest possible value of xy is 20.

So, the correct answer is b. 20.