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\n" ); document.write( "Let c be a real number. What is the maximum value of c such that the graph of the
\n" ); document.write( "parabola y = 1/5 x^2 has at most one point of intersection with the line y = cx?
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document.write( " Let' first determine the set of possible values of \"c\", such that the graph \r\n" );
document.write( " of the parabola y = 1/5 x^2 has at most one point of intersection with the line y = cx?\r\n" );
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document.write( "This parabola and all such lines ALWAYS have at least one intersection (or common) point: \r\n" );
document.write( "this point is (0,0), the origin of the coordinate system.\r\n" );
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document.write( "Next, if c =/= 0 (i.e. if the straight line has non-zero slope), then certainly\r\n" );
document.write( "there is another intersection point.\r\n" );
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document.write( "This fact is obvious, if to recall that our parabola (as any other parabola)\r\n" );
document.write( "raises faster than any non-degenerated linear function that has common point \r\n" );
document.write( "with the parabola.- so, another intersection point does exist inevitably.\r\n" );
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document.write( "From this reasoning, you see that there is only one straight line of the form y= cx\r\n" );
document.write( "which has only one intersection point with the given parabola.\r\n" );
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document.write( "This line is the horizontal line y = 0 with c = 0.\r\n" );
document.write( "The intersection point is the tangent point (0,0), the origin of the coordinate system.\r\n" );
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document.write( "Thus the set of all possible coefficients {c} consists of one single element c= 0.\r\n" );
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document.write( "Therefore, the ANSWER to the problem's question is THIS\r\n" );
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document.write( " +---------------------------------------------------------------+\r\n" );
document.write( " | the maximum value of c such that the graph of the |\r\n" );
document.write( " | parabola y = 1/5 x^2 has at most one point of intersection |\r\n" );
document.write( " | with the line y = cx is 0 (zero). |\r\n" );
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document.write( "Again, the set of all such values \"c\" consists of one single element 0 (zero),\r\n" );
document.write( "and the maximum value of all such \"c\"s is 0, naturally.\r\n" );
document.write( "\r\n" ); document.write( "\n" ); document.write( "Solved and completed, with all necessary explanations.\r
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