Suppose that for all reals 0 ≤ a ≤ b ≤ c ≤ d , we have :
(a + b + c + d)^2 ≥ K*b*c
Find the largest possible value of K.
If a=0 then the left side is as small as possible without affecting the right side. Similarly if c=d then the left side is as small as possible without affecting the right side.
So the limiting case is a=0, c=d making the equation (b+2c)^2 >= K*bc.
If I apply the arithmetic-geometric inequality to b and 2c I get (b+2c)/2 >= sqrt(2bc). Multiply by 2 and square to get (b+2c)^2 >= 8bc. So then k is at least 8, but the am-gm equality only holds when b=2c But we don't have that. here since b<=c.
The closest we can get to the equality condition is b=c. So make that substitution. Then (3b)^2=K*b^2. Then K=9 is the value of K we are to find.
Solution
