X and y are positive integers and x+y=11. What is the largest possible value of 1/x -1/y?

x = 1 and y = 10 -->1-.1 = .9

try others
x = 2 and y =9 --> 1/2 - 1/9
see?

Using Calculus ....

x+y = 11 ---> y = 11-x

S = 1/x - 1/y
= 1/x - 1/(11-x) = x^-1 - (11-x)^-1
dS/dx = -1/x^2 - 1/(11-x)^2
= 0 for a max/min of S

-1/(11-x)^2 = 1/x^2
-x^2 = 11-x^2
-2x^2 = 11
x^2 = -11/2
No real solution , there is no maximum
see Wolfram
http://www.wolframalpha.com/input/?i=plot+y+%3D+1%2Fx+-+1%2F%2811-x%29

1/x - 1/(11-x) becomes infinitely large as x ---> 0

To find the largest possible value of 1/x - 1/y, we need to find the largest value of x and the smallest value of y that satisfy the given equation x + y = 11.

Let's consider the extreme cases where x is the largest and y is the smallest.

Case 1: x = 10, y = 1
1/x - 1/y = 1/10 - 1/1 = 1/10 - 1 = -9/10

Case 2: x = 9, y = 2
1/x - 1/y = 1/9 - 1/2 = 2/18 - 9/18 = -7/18

Case 3: x = 8, y = 3
1/x - 1/y = 1/8 - 1/3 = 3/24 - 8/24 = -5/24

Case 4: x = 7, y = 4
1/x - 1/y = 1/7 - 1/4 = 4/28 - 7/28 = -3/28

From these cases, we can see that as x decreases and y increases, the value of 1/x - 1/y becomes smaller and smaller. Therefore, the largest possible value of 1/x - 1/y occurs when x is the smallest and y is the largest.

Case 5: x = 1, y = 10
1/x - 1/y = 1/1 - 1/10 = 10/10 - 1/10 = 9/10

Therefore, the largest possible value of 1/x - 1/y is 9/10.

To find the largest possible value of 1/x - 1/y, we need to find the smallest possible value of y and the largest possible value of x.

Given that x + y = 11, we can start by trying different values of y and calculating the corresponding value of x:

If y = 1, then x = 11 - 1 = 10, and 1/x - 1/y = 1/10 - 1 = -0.9.

If y = 2, then x = 11 - 2 = 9, and 1/x - 1/y = 1/9 - 1/2 = -0.778.

If y = 3, then x = 11 - 3 = 8, and 1/x - 1/y = 1/8 - 1/3 = +0.042.

If y = 4, then x = 11 - 4 = 7, and 1/x - 1/y = 1/7 - 1/4 = +0.089.

If y = 5, then x = 11 - 5 = 6, and 1/x - 1/y = 1/6 - 1/5 = +0.083.

If y = 6, then x = 11 - 6 = 5, and 1/x - 1/y = 1/5 - 1/6 = +0.033.

If y = 7, then x = 11 - 7 = 4, and 1/x - 1/y = 1/4 - 1/7 = +0.089.

If y = 8, then x = 11 - 8 = 3, and 1/x - 1/y = 1/3 - 1/8 = +0.208.

If y = 9, then x = 11 - 9 = 2, and 1/x - 1/y = 1/2 - 1/9 = +0.361.

If y = 10, then x = 11 - 10 = 1, and 1/x - 1/y = 1/1 - 1/10 = +0.9.

From the calculations above, we can see that the largest possible value of 1/x - 1/y is 0.9, which occurs when y = 1 and x = 11 - 1 = 10.