Let x, y be non-negative numbers with
sqrt(1-x^2/4)+sqrt(1-y^2/16)=3/2.
Find the maximum value of xy.
Maximizing xy will also maximize 2xy. Then the given conditional equation can be rewritten as sqrt(1-(2x)^2/16) + sqrt(1-y^2/16) = 3/2.
Now the equation is symmetric in 2x and y. I'll take that (2x)y is maximized when sqrt(1-(2x)^2/16) and sqrt(1-y^2/16) are each each equal to 3/4. (Arguing since 2x and y are symmetric that the equation will 'balance' when (2x)*y is maximized)
Solving sqrt(1-y^2/16) = 3/4 gives y=sqrt(7), then x=sqrt(7)/2, implying the maximum value of xy is 7/2.
Edited on December 24, 2021, 8:47 am
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