document.write( "Question 1111168: Describe the behavior of the curve y = - x3 + 2x2 + 4x - 5 at the point (1, 0).\r
\n" ); document.write( "\n" ); document.write( "Select all that apply.\r
\n" ); document.write( "\n" ); document.write( " increasing
\n" ); document.write( " decreasing
\n" ); document.write( " concave up
\n" ); document.write( " concave down \n" ); document.write( "
Algebra.Com's Answer #726192 by KMST(5330)\"\" \"About 
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The function's derivative, \"%22y+%27%22\" or \"dy%2Fdx\" is
\n" ); document.write( "\"dy%2Fdx=-3x%5E2%2B4x%2B4\" or \"%22y+%27%22=-3x%5E2%2B4x%2B4\" .
\n" ); document.write( "When \"x=1\" , \"y\" is indeed zero:
\n" ); document.write( "\"y+=+-+1%5E3+%2B+2%2A1%5E2+%2B+4%2A1+-+5=-1%2B2%2B4-5=0\" ,
\n" ); document.write( "so the graph does go through (1,0).
\n" ); document.write( "When \"x=0\" , the derivative's value is
\n" ); document.write( "\"-3%2A1%5E2%2B4%2A1%2B4=-3%2B4%2B4=5\" ,
\n" ); document.write( "so \"5\" is the slope of the tangent to the graph at (1,0) .
\n" ); document.write( "That means \"y\" is \"highlight%28increasing%29\" , and increasing steeply.
\n" ); document.write( "Obviously, if \"y\" is increasing around \"x=1\",
\n" ); document.write( "it is NOT decreasing.
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\n" ); document.write( "The second derivative, \"%22y+%27+%27%22\" or \"d%5E2y%2Fdx%5E2\" ,
\n" ); document.write( "is the derivative of \"%22y+%27%22=-3x%5E2%2B4x%2B4\" ;
\n" ); document.write( "\"%22y+%27+%27%22=-6x%2B4\" .
\n" ); document.write( "For \"x=1\" , the value of that second derivative is \"-6%2A1%2B4=-6%2B4=-2\" .
\n" ); document.write( "The fact that that value is negative means that \"%22y+%27%22\" is decreasing,
\n" ); document.write( "which means that the slope of the curve is increasing,
\n" ); document.write( "so it is growth rate is slowing,
\n" ); document.write( "and the curve is curling down.
\n" ); document.write( "It is \"concave\" \"highlight%28down%29\" , like a frown.
\n" ); document.write( "If the graph of \"y=-x%5E3%2B2x%5E2%2B4x-5\" is concave down,
\n" ); document.write( "it is obviously NOT concave up.
\n" ); document.write( "
\n" ); document.write( "The \"red%28graph%29\" of \"y=-x%5E3%2B2x%5E2%2B4x-5\" ,
\n" ); document.write( "and the \"green%28tangent%29\" to that curve at (1,0) are shown below.
\n" ); document.write( "\"graph%28300%2C300%2C-4%2C6%2C-6.5%2C3.5%2C-x%5E3%2B2x%5E2%2B4x-5%2C5x-137%2F27%29\"
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\n" ); document.write( "The graph above (and the one you could get in a graphing calculator),
\n" ); document.write( "shows that \"y\" is increasing at (1,0).
\n" ); document.write( "Because the tangent slope is so steep, not large,
\n" ); document.write( "it is not visually obvious that the curve is concave down,
\n" ); document.write( "although changing the scale and zooming helps:
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