document.write( "Question 816619: Please help me solve this equation:\r
\n" ); document.write( "\n" ); document.write( "Find all asymptotes, intercepts and graph. you must also find all the critical points and inflections points (if any)\r
\n" ); document.write( "\n" ); document.write( "f(x) = x^3-3x^2-10x/x^2+5x+6 \n" ); document.write( "
Algebra.Com's Answer #491680 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "f(x) = \"%28x%5E3-3x%5E2-10x%29%2F%28x%5E2%2B5x%2B6%29\"\r\n" );
document.write( "\r\n" );
document.write( "Factor the numerator:     Factor the denominator\r\n" );
document.write( "\r\n" );
document.write( "    x³-3x²-10x                x²+5x+6                    \r\n" );
document.write( "    x(x²-3x-10)              (x+3)(x+2)\r\n" );
document.write( "    x(x+3)(x+2)\r\n" );
document.write( "\r\n" );
document.write( "f(x) = \"%28x%28x-5%29%28x%2B2%29%29%2F%28%28x%2B3%29%28x%2B2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Since (x+3) is a factor of the denominator but not\r\n" );
document.write( "the numerator, there is an asymptote where x+3=0,\r\n" );
document.write( "or at x=-3, which is the equation of the vertical\r\n" );
document.write( "asymptote, where there is a non-removable discontinuite.\r\n" );
document.write( " Since (x+2) is a factor of both denominator and numerator,\r\n" );
document.write( "there is a removable discontinuity where x+2=0, at x=-2.\r\n" );
document.write( "\r\n" );
document.write( "We may cancel the (x+2)'s as long as we also state that\r\n" );
document.write( "x≠2\r\n" );
document.write( "\r\n" );
document.write( "f(x) = \"x%28x-5%29%2F%28x%2B3%29\", where x≠2\r\n" );
document.write( "\r\n" );
document.write( "So we graph\r\n" );
document.write( "\r\n" );
document.write( "y = \"x%28x-5%29%2F%28x%2B3%29\", leaving a hole at x=2\r\n" );
document.write( "\r\n" );
document.write( "There is a vertical asymptote at x=-3\r\n" );
document.write( "Since the degree of the numberator is 1 more than the degree\r\n" );
document.write( "of the denominator, there is no horizontal asymptote, but there\r\n" );
document.write( "is an oblique (or slant) asymptote, which we find by long\r\n" );
document.write( "division:\r\n" );
document.write( "\r\n" );
document.write( "We have to multiply the numerator out and add +0 to divide:\r\n" );
document.write( "\r\n" );
document.write( "y = \"x%B2-5x%2B0%29%2F%28x%2B3%29\",\r\n" );
document.write( "\r\n" );
document.write( "        x- 8+\"34%2F%28x%2B3%29\"\r\n" );
document.write( "x+3)x²-5x+ 0\r\n" );
document.write( "    x²+3x\r\n" );
document.write( "      -8x+ 0\r\n" );
document.write( "      -8x-24\r\n" );
document.write( "          34\r\n" );
document.write( "\r\n" );
document.write( "Since the fraction \"34%2F%28x%2B3%29\" approaches 0 as x gets large,\r\n" );
document.write( "the graph of f(x) must approach the line y=x-8, which is the\r\n" );
document.write( "equation of the oblique (slant) asymptote.\r\n" );
document.write( "\r\n" );
document.write( "We get the y-intercept by setting x = 0\r\n" );
document.write( "\r\n" );
document.write( "y = \"0%280-5%29%2F%280%2B3%29\" = 0\r\n" );
document.write( "\r\n" );
document.write( "So the y-intercept is (0,0)\r\n" );
document.write( "\r\n" );
document.write( "We get the x-intercepts by setting y = 0 \r\n" );
document.write( "\r\n" );
document.write( "0 = \"x%28x-5%29%2F%28x%2B3%29\"\r\n" );
document.write( "\r\n" );
document.write( "0 = x(x-5)\r\n" );
document.write( "    x=0;  x-5=0\r\n" );
document.write( "            x=5\r\n" );
document.write( "\r\n" );
document.write( "So the x-intercepts are (0,0) and (5,0)\r\n" );
document.write( "\r\n" );
document.write( "We plot the asymptotes and the intercepts:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Now we find any relative extrema points by\r\n" );
document.write( "finding the derivative and setting it = 0\r\n" );
document.write( "\r\n" );
document.write( "y = \"%28x%28x-5%29%29%2F%28x%2B3%29\"\r\n" );
document.write( "Multiply the top out:\r\n" );
document.write( "y = \"%28x%5E2-5x%29%2F%28x%2B3%29\"\r\n" );
document.write( "Use the quoptient formula for the derivative:\r\n" );
document.write( "y' = \"%28%28x%2B3%29%282x-5%29-%28x%5E2-5x%29%281%29%29%2F%28x%2B3%29%5E2\"\r\n" );
document.write( "y' = \"%282x%5E2-5x%2B6x-15-x%5E2%2B5x%29%2F%28x%2B3%29%5E2\"\r\n" );
document.write( "y' = \"%28x%5E2%2B6x-15%29%2F%28x%2B3%29%5E2\"\r\n" );
document.write( "Setting that = 0 to find relative extrema:\r\n" );
document.write( "\"%28x%5E2%2B6x-15%29%2F%28x%2B3%29%5E2\" = 0\r\n" );
document.write( "x²+6x-15 = 0\r\n" );
document.write( "Unfortunately that doesn't factor, so we must\r\n" );
document.write( "use the quadratic formula:\r\n" );
document.write( "\"x+=+%28-6+%2B-+sqrt%28+6%5E2-4%2A1%2A%28-15%29+%29%29%2F%282%2A1%29+\"\r\n" );
document.write( "\"x+=+%28-6+%2B-+sqrt%2836%2B60%29%29%2F2+\"\r\n" );
document.write( "\"x+=+%28-6+%2B-+sqrt%2896%29%29%2F2+\"\r\n" );
document.write( "\"x+=+%28-6+%2B-+sqrt%2816%2A6%29%29%2F2+\"\r\n" );
document.write( "\"x+=+%28-6+%2B-+4sqrt%286%29%29%2F2+\"\r\n" );
document.write( "\"x+=+%282%28-3+%2B-+2sqrt%286%29%29%29%2F2+\"\r\n" );
document.write( "\"x+=+%28cross%282%29%28-3+%2B-+2sqrt%286%29%29%29%2Fcross%282%29+\"\r\n" );
document.write( "x = -3 ± 2V6\r\n" );
document.write( "Approximating:  x=-7.90 and x=1.90\r\n" );
document.write( "Substuting those in y, we get approximately\r\n" );
document.write( "                y=-20.8 and y=-1.20\r\n" );
document.write( "\r\n" );
document.write( "Relative extrema candidates are approximately (-7.90,-20.8)\r\n" );
document.write( "and (1.90,-1.20)\r\n" );
document.write( "\r\n" );
document.write( "To find out whether they are relative maximums or minimums,\r\n" );
document.write( "or any inflection points, we must find the second derivative:\r\n" );
document.write( "\r\n" );
document.write( "y' = \"%28x%5E2%2B6x-15%29%2F%28x%2B3%29%5E2\"\r\n" );
document.write( "Use the quotient formula:\r\n" );
document.write( "y\" = \"%28%28x%2B3%29%5E2%282x%2B6%29-%28x%5E2%2B6x-15%292%28x%2B3%29%281%29%29%2F%28x%2B3%29%5E4\"\r\n" );
document.write( "y\" = \"%28%28x%5E2%2B6x%2B9%29%282x%2B6%29-%28x%5E2%2B6x-15%29%282x%2B6%29%29%2F%28x%2B3%29%5E4\"\r\n" );
document.write( "y\" = \"%28%282x%5E3%2B18x%5E2%2B54x%2B54%29-%282x%5E3%2B18x%5E2%2B6x-90%29%29%2F%28x%2B3%29%5E4\"  \r\n" );
document.write( "y\" = \"%282x%5E3%2B18x%5E2%2B54x%2B54-2x%5E3-18x%5E2-6x%2B90%29%2F%28x%2B3%29%5E4\"\r\n" );
document.write( "y\" = \"%2848x%2B144%29%2F%28x%2B3%29%5E4\"\r\n" );
document.write( "y\" = \"%2848%28x%2B3%29%29%2F%28x%2B3%29%5E4\"\r\n" );
document.write( "y\" = \"48%2F%28x%2B3%29%5E3\"\r\n" );
document.write( "\r\n" );
document.write( "Substituting x=-7.90, y\" comes out negative,\r\n" );
document.write( "therefore the point (-7.90,-20.8) is a relative\r\n" );
document.write( "maximum, since the curvature is downward\r\n" );
document.write( "\r\n" );
document.write( "Substituting x=1.90, y\" comes out positive,\r\n" );
document.write( "therefore the point (1.90,-1.20) is a relative\r\n" );
document.write( "minimum, since the curvature is upward\r\n" );
document.write( " \r\n" );
document.write( "To find any inflection points we set y\"=0\r\n" );
document.write( "\r\n" );
document.write( "\"48%2F%28x%2B3%29%5E3\" = 0\r\n" );
document.write( "48 = 0\r\n" );
document.write( "A contradiction so there are no inflection points.\r\n" );
document.write( "\r\n" );
document.write( "So we draw the graph:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "What a terribly long and messy problem!\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );