document.write( "Question 1207468: Suppose that f(x) and g(x) are functions such that the range of f is [-5,3], and the range of g is [-2,1]. The range of f(x) * g(x) is [a,b]. What is the largest possible value of b? \n" ); document.write( "

Algebra.Com's Answer #845344 by ikleyn(52798) $\"\"$ $\"About$

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\n" ); document.write( "Suppose that f(x) and g(x) are functions such that the range of f is [-5,3], and the range of g is [-2,1].
\n" ); document.write( "The range of f(x) * g(x) is [a,b]. What is the largest possible value of b?
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document.write( "From one side, we have this inequality\r\n" );
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document.write( "    f(x) * g(x) <= |f(x)| * |g(x)|.\r\n" );
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document.write( "It tells us that b <= max(|f(x)| * |g(x)|) <= max|f(x)| * max|g(x)| = 5*2 = 10.    (1)\r\n" );
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document.write( "From the other side,    in some point x,  f(x) = -5, g(x) = -2  ,    f(x)*g(x) = (-5)*(-2) = 10.\r\n" );
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document.write( "Therefore, then b >= 10.    (2)\r\n" );
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document.write( "From these two inequalities, (1) and (2), we conclude that the maximum possible value of b is  (-5)*(-2) = 10.\r\n" );
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document.write( "        It does not mean necessary that max (f(x)*g(x)) is always 10,  \r\n" );
document.write( "        for any given functions f(x) and g(x), satisfying the condition.\r\n" );
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document.write( "What it means precisely in this problem, are two facts:\r\n" );
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document.write( "    (a)  b can not be greater than 10;  \r\n" );
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document.write( "and\r\n" );
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document.write( "    (b)  for some functions f(x) and g(x) under given conditions, \"b\" can be 10.\r\n" );
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