document.write( "Question 1193060: The positive variables x and y are such that x^4y=32. A third variable z is defined by z = x^2 + y
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Algebra.Com's Answer #825055 by ikleyn(52797) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( " =        (1)\r\n" );
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document.write( " =             (2)\r\n" );
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document.write( " =             (3)\r\n" );
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document.write( "The stationary point is where the derivative is zero.\r\n" );
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document.write( " = \r\n" );
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document.write( " = \r\n" );
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document.write( "x = 2    (actually, x = +/- 2, but since we consider everything in positive numbers, we take x = 2).\r\n" );
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document.write( "At the stationary point,  and  =  =  = \r\n" );
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document.write( "    The stationary point is a minimum if the second derivative at the point is positive; \r\n" );
document.write( "    or it is a maximum if that derivative is negative.  \r\n" );
document.write( "    At x = 2, the second derivative is OBVIOULSLY positive (it is clear without any calculations)\r\n" );
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document.write( "So the stationary point is a minimum.\r\n" );
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document.write( "ANSWER: z has a stationary point that is a minimum when x = 2 and y = 2.\r\n" );
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document.write( "To make this result visually verifiable, I prepared a plot below.\r\n" );
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document.write( "            Plot  z =  +   (see formula (1)\r\n" );
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