how to solve this problem:
max min(x-y)..... here where the max have written, under it x>=0 is also given. also under min y>=0 is given.
Also tell that is this is equals to min max(x-y) ? (same as above under max x>=0, under min y>=0 is given)
x and y both in R
To solve the problem of maximizing the minimum value of (x - y), where x and y are both in the set of real numbers (R), under the given constraints (x >= 0 and y >= 0), you can follow these steps:
1. Start by considering the case when x = 0:
- If x = 0, then the equation becomes (0 - y) = -y.
- Since y is non-negative (y >= 0), the minimum value of (x - y) is -y, which is equal to 0 when y = 0.
2. Next, consider the case when x > 0:
- If x > 0, let's assume y is the minimum value. So, y = 0.
- In this case, (x - y) = (x - 0) = x.
- Since x is non-negative (x >= 0), the maximum value of (x - y) is x.
Therefore, the maximum value of (x - y) is x, under the given constraints x >= 0 and y >= 0.
As for the question of whether min max(x-y) is equal to max min(x-y), let's analyze it step by step:
1. min max(x-y):
- This means finding the minimum value of the maximum value of (x - y) under the given constraints.
- Since we've already determined that the maximum value of (x - y) is x, we need to find the minimum value of x.
- The given constraint x >= 0 indicates that there is no lower bound for x, so the minimum value of x does not exist.
2. max min(x-y):
- This means finding the maximum value of the minimum value of (x - y) under the given constraints.
- We've already determined that the minimum value of (x - y) is -y.
- The given constraint y >= 0 indicates that the minimum value of y is 0.
- Therefore, the maximum value of (x - y) is (0 - 0) = 0.
In conclusion, min max(x-y) does not exist, whereas max min(x-y) is equal to 0 under the given constraints.
To solve the problem "max min(x-y)" where x>=0 is given under the max and y>=0 is given under the min, we need to find the maximum value of the minimum of (x-y) given the specified conditions.
First, let's analyze the given conditions for x and y:
- x>=0: This means that x can take any non-negative real number.
- y>=0: Similarly, this means that y can take any non-negative real number.
Now, let's proceed to solve the problem:
1. To find the minimum value of (x-y), we need to consider all possible combinations of x and y.
2. Start by assuming the minimum value for x and y as 0.
3. Substitute these values into the expression (x-y) to get the minimum value of (x-y), which is 0 - 0 = 0.
4. Next, we need to consider the possibility of x and y being greater than 0.
5. Since we are looking for the minimum value of the expression x-y under the given conditions, let's consider the scenario where x and y both take the value 0.
- In this case, the expression (x-y) will be 0 - 0 = 0.
6. Now, consider the scenario where x is any positive number greater than 0 and y is 0.
- In this case, the expression (x-y) will be x - 0 = x.
7. Finally, consider the scenario where x is 0 and y is any positive number greater than 0.
- In this case, the expression (x-y) will be 0 - y = -y.
To summarize the above analysis, the minimum value of (x-y) is 0 under the given conditions.
Now, let's move on to the question of whether "max min(x-y)" is equal to "min max(x-y)".
When we have "max min(x-y)", it means we first take the minimum value of (x-y) and then find the maximum value of that minimum.
On the other hand, when we have "min max(x-y)", it means we first take the maximum value of (x-y) and then find the minimum value of that maximum.
These two expressions are not necessarily equal. The order of operations matters in this case. So, "max min(x-y)" is different from "min max(x-y)".
In summary:
- The solution to "max min(x-y)" where x>=0 and y>=0 is given is 0.