I have the following equations and inequalities: 1 = A ′ + B ′ 1...

Carly Cannon

Carly Cannon

Answered

2022-07-02

I have the following equations and inequalities:
1 = A + B
1 = A + B + C
A A
B B
All variables are bounded below by zero and above by one.
I wonder if I can find an analytic expression for the upper and lower bounds for the difference A ( 1 C ) A

Answer & Explanation

lofoptiformfp

lofoptiformfp

Expert

2022-07-03Added 16 answers

Step 1
First note A ( 1 C ) A = A ( A + B ) A
For the upper bound clearly we need to take A = 0 , B = B and C = A . In which case the problem comes down to finding the maximum value of A B = A A 2 which is obtained if A = 1 2 so an upper bound is 1 4 . For a lower bound clearly we need to take A = A , B = 0 and C = B , so we have to find the minimum value of A 2 A for 0 A 1 , which is obtained if A = 1 2 . So a lower bound is 1 4 .
So 1 4 A ( 1 C ) A 1 4 .

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