document.write( "Question 1171194: Verify that the hypotheses of the Mean Value Theorem are satisfied on the given interval and find all values c that satisfies the conclusion of the theorem if g(x) = x + 1/x, 3 =< x =< 4. \n" ); document.write( "

Algebra.Com's Answer #796113 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "    In calculus, the Extreme Value Theorem states that if a real-valued function f is continuous on \r\n" );
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document.write( "    the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. \r\n" );
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\n" ); document.write( "\n" ); document.write( "See this Wikipedia article
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document.write( "So, the function is assumed to be CONTINUOUS.\r\n" );
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document.write( "The given function is the sum of two elementary functions,  x  and  .\r\n" );
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document.write( "Each of the two functions is continues on the given interval, as it is known from Calculus.\r\n" );
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document.write( "Therefore, the sum of these functions is continue on the given interval.\r\n" );
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document.write( "It proves the statement.\r\n" );
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document.write( "To answer the last question, you need to solve this quadratic equation \r\n" );
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document.write( "    x + 1/x = c.\r\n" );
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document.write( "Reduce it to the standard form\r\n" );
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document.write( "    x^2 + 1 = cx\r\n" );
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document.write( "    c^2 - cx + 1 = 0\r\n" );
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document.write( "Calculate the discriminant\r\n" );
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document.write( "    d = .\r\n" );
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document.write( "The equation is solvable in real numbers if and only if  d >= 0, which is\r\n" );
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document.write( "    c^2 >= 4,\r\n" );
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document.write( "giving\r\n" );
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document.write( "    c <= -2   OR   c >= 2.     (1)\r\n" );
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document.write( "Since our function g(x) = x +   is always positive on the given interval,  \r\n" );
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document.write( "the inequality (1)  for \"c\" is reduced to\r\n" );
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document.write( "    c >= 2.\r\n" );
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document.write( "The plot below illustrate the function\r\n" );
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document.write( "    \r\n" );
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document.write( "            Plot y = x + 1/x\r\n" );
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document.write( "Next, to get the final description for the range of \"c\"-values, we should evaluate the function at the ends of the interval\r\n" );
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document.write( "    g(3) =  = 1 ;  g(4) =  = 4 .\r\n" );
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document.write( "Now we select the smallest and the greatest values of these two end-points, which are  3  and  4 .\r\n" );
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document.write( "In this way, we obtain inequalities for \"c\" in the FINAL form\r\n" );
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document.write( "        3  <= c <= 4 .\r\n" );
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