document.write( "Question 343970: If a = y^2 and b = y^4, and -1 ≤ y ≤ 3, then what is the largest possible difference between a and b?\r
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Algebra.Com's Answer #246083 by Edwin McCravy(20065) $\"\"$ $\"About$

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If a = y^2 and b = y^4, and -1 ≤ y ≤ 3, then what is the largest possible difference between a and b?
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document.write( "Maximum values of continuous functions occur at values of the\r\n" );
document.write( "independent variable where the derivative is 0 or at\r\n" );
document.write( "endpoints of interval on which the variable is defined.\r\n" );
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document.write( "The difference is largest when  is the largest.\r\n" );
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document.write( "Let this difference be z.\r\n" );
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document.write( "Maximum values of continuous functions occur at values of the\r\n" );
document.write( "independent variable (in this cases y) where the derivative is 0 or at\r\n" );
document.write( "endpoints of interval on which the variable is defined.\r\n" );
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document.write( "We set that = 0\r\n" );
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document.write( "  \r\n" );
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document.write( "Using the 2nd derivative test:\r\n" );
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document.write( "Substituting y=0 gives \r\n" );
document.write( "which is positive so this gives a relative minimum value,\r\n" );
document.write( "which is not the value we want\r\n" );
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document.write( "Substituting   gives \r\n" );
document.write( "which is negative so this value of y gives a relative maximum value.\r\n" );
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document.write( "To find this relative maximum value we substitute \r\n" );
document.write( "in the original equation \r\n" );
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document.write( "This is a candidate for the correct answer, unless the endpoints\r\n" );
document.write( "of the given interval give a larger value:\r\n" );
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document.write( "The end points of the given interval are -1 and 3\r\n" );
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document.write( "Substituting the endpoint -1\r\n" );
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document.write( "That's not larger than !\r\n" );
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document.write( "Substituting the endpoint 3\r\n" );
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document.write( "That's certainly not larger than !\r\n" );
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document.write( "Therefore the maximum difference between  and  is \r\n" );
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document.write( "Edwin

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