Maximize $(ab+cd)^2$
$begingroup$
$a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$
$ac + bd = 240$ where a, b ,c ,d are positive reals, maximize $(ab +cd)^2$
looking at the equations and restrictions, I think Cauchy can be applied
$(a^2+c^2)(b^2+d^2)> (ab + cd)^2$ so we just need to find the value of the LHS
but I'm finding it hard to manipulate them and i cant seem to find an application for the ac+bd =240.
optimization
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
$endgroup$
add a comment |
$begingroup$
$a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$
$ac + bd = 240$ where a, b ,c ,d are positive reals, maximize $(ab +cd)^2$
looking at the equations and restrictions, I think Cauchy can be applied
$(a^2+c^2)(b^2+d^2)> (ab + cd)^2$ so we just need to find the value of the LHS
but I'm finding it hard to manipulate them and i cant seem to find an application for the ac+bd =240.
optimization
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
$endgroup$
$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02
add a comment |
$begingroup$
$a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$
$ac + bd = 240$ where a, b ,c ,d are positive reals, maximize $(ab +cd)^2$
looking at the equations and restrictions, I think Cauchy can be applied
$(a^2+c^2)(b^2+d^2)> (ab + cd)^2$ so we just need to find the value of the LHS
but I'm finding it hard to manipulate them and i cant seem to find an application for the ac+bd =240.
optimization
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
$endgroup$
$a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$
$ac + bd = 240$ where a, b ,c ,d are positive reals, maximize $(ab +cd)^2$
looking at the equations and restrictions, I think Cauchy can be applied
$(a^2+c^2)(b^2+d^2)> (ab + cd)^2$ so we just need to find the value of the LHS
but I'm finding it hard to manipulate them and i cant seem to find an application for the ac+bd =240.
optimization
optimization
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
edited Nov 30 '18 at 10:01
SuperMage1
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
asked Nov 28 '18 at 13:19
SuperMage1SuperMage1
877210
877210
$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02
add a comment |
$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02
$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02
add a comment |
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$begingroup$
What are the constraints?$ a^2 + b^2 - frac{ab}{2} = c^2 + d^2 + frac{cd}{2} = 256$?
$endgroup$
– gimusi
Nov 28 '18 at 13:28
$begingroup$
and ac + bd = 240...
$endgroup$
– SuperMage1
Nov 28 '18 at 13:33
$begingroup$
sorry, edited already
$endgroup$
– SuperMage1
Nov 30 '18 at 9:34
$begingroup$
last edit, its correct now.
$endgroup$
– SuperMage1
Nov 30 '18 at 10:02