document.write( "Question 821307: x^2-3x^5>10 \n" ); document.write( "

Algebra.Com's Answer #494094 by KMST(5328) $\"\"$ $\"About$

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$\"x%5E2-3x%5E5%3E10\"$
\n" ); document.write( " $\"-3x%5E5%2Bx%5E2-10%3E0\"$
\n" ); document.write( " $\"3x%5E5-x%5E2%2B10%3C0\"$
\n" ); document.write( "A graphing calculator would tell you that
\n" ); document.write( " $\"f%28x%29=3x%5E5-x%5E2%2B10=0\"$ happens for $\"x=-1.2311\"$ (rounded),
\n" ); document.write( "with the $\"3x%5E5-x%5E2%2B10%3C0\"$ only for $\"x%3C-1.2311\"$ .
\n" ); document.write( "Before graphing calculators, I would have calculated the derivative as
\n" ); document.write( " $\"df%2Fdx=15x%5E4-2x=15x%28x%5E3-2%2F15%29\"$ =

\n" ); document.write( "Since the last factor is always positive, the zeros of the derivative are at $\"x=0\"$ and at $\"x=root%283%2C2%2F15%29\"$ .
\n" ); document.write( "The derivative is negative in between those two values of $\"x\"$ ,
\n" ); document.write( "meaning that is an interval where $\"f%28x%29\"$ decreases.
\n" ); document.write( "For other values of $\"x\"$ , the derivative is positive and the function increases.
\n" ); document.write( "At $\"x=0\"$ where $\"f%280%29=10\"$ we have a maximum of $\"f%28x%29\"$ ,
\n" ); document.write( "and at $\"x=root%283%2C2%2F15%29\"$ , a minimum of $\"f%28x%29\"$ , where $\"f%28x%29%3E0\"$ .
\n" ); document.write( "Since $\"f%28-2%29=-90\"$ and $\"f%28-1%29=6\"$ ,
\n" ); document.write( "The value of $\"x\"$ that makes $\"f%28x%29=0\"$ is in between, $\"-2%3Cx%3C-1\"$ .
\n" ); document.write( "At that point we would try guess-and-check values in between, aiming to get closer limits on $\"x\"$ , maybe going through
\n" ); document.write( " $\"-1.5%3Cx%3C-1\"$ , $\"-1.5%3Cx%3C-1.2\"$ , $\"-1.3%3Cx%3C-1\"$ and so on, until I got a close enough approximation. \n" ); document.write( "

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