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Find the largest constant \(c\) so that

\(x^2 + y^2 + 1 \ge c(x + y)\)

for all real numbers \(x\) and \(y\)

 

pls i need it done quick

 Jan 7, 2021
edited by Guest  Jan 7, 2021
 #1
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By Lagrange Multipliers, you get the optimal value when x = y = 1:

x^2 + y^2 + 1 >= c(x + y)

3 >= 2c

c <= 3/2

 

So the largest constant c is 3/2.

 Jan 7, 2021
 #2
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For those of us who are unsure, are negative numbers real numbers ?

If that's the case, then if both x and y are negative, c can be as large as you please.

 Jan 7, 2021

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