document.write( "Question 1027081: Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph. \n" ); document.write( "

Algebra.Com's Answer #642350 by Fombitz(32388) $\"\"$ $\"About$

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$\"Z=xy\"$
\n" ); document.write( "To simplify, the product of $\"Z%5E2\"$ would also be a maximum so find the maximum value of $\"Z%5E2=%28x%5E2y%5E2%29\"$ .
\n" ); document.write( "From the equation,
\n" ); document.write( " $\"x%5E2=16-2y%5E2%29\"$
\n" ); document.write( "Substituting,
\n" ); document.write( " $\"Z%5E2=%2816-2y%5E2%29y%5E2\"$
\n" ); document.write( " $\"g%28y%29=Z%5E2=16y%5E2-2y%5E4\"$
\n" ); document.write( "Find the derivative of $\"g=Z%5E2\"$ ,
\n" ); document.write( " $\"dg%2Fdy=32y-8y%5E3\"$
\n" ); document.write( " $\"32y-8y%5E3=0\"$
\n" ); document.write( " $\"8y%284-y%5E2%29=0\"$
\n" ); document.write( "Two possible solutions:
\n" ); document.write( " $\"y=0\"$
\n" ); document.write( "This is not a maximum but a local minimum (check with second derivative test).
\n" ); document.write( " $\"4-y%5E2=0\"$
\n" ); document.write( " $\"y%5E2=4\"$
\n" ); document.write( " $\"y=0+%2B-+2\"$
\n" ); document.write( "Then,
\n" ); document.write( " $\"x%5E2=16-2%282%29%5E2\"$
\n" ); document.write( " $\"x%5E2=8\"$
\n" ); document.write( " $\"x=0+%2B-+2sqrt%282%29\"$
\n" ); document.write( "So the maximum value would be,
\n" ); document.write( " $\"xy=2%2A2%28sqrt%282%29%29=4sqrt%282%29\"$ \r
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