document.write( "Question 1094380: How do i determine the intervals where the function is increasing and where it is decreasing in
\n" ); document.write( "f(x) = 2x^3 + 2x^2 - 2x + 2 \n" ); document.write( "
Algebra.Com's Answer #708950 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Graphs change from increasing to decreasing at points where the\r\n" );
document.write( "tangent line is horizontal (has slope 0)\r\n" );
document.write( "\r\n" );
document.write( "So we find the points on the graph where the tangent line is \r\n" );
document.write( "horizontal.  The derivative is a formula for the slope of a\r\n" );
document.write( "tangent line, so we find the derivative and set it equal to\r\n" );
document.write( "0, and solve for the values of x when the graph has a \r\n" );
document.write( "horizontal tangent line: \r\n" );
document.write( "\r\n" );
document.write( "f(x) = 2x³ + 2x² - 2x + 2\r\n" );
document.write( "\r\n" );
document.write( "f'(x) = 6x² + 4x - 2        <--derivative, set = 0\r\n" );
document.write( "\r\n" );
document.write( "6x² + 4x - 2 = 0\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 2\r\n" );
document.write( "\r\n" );
document.write( "3x² + 2x - 1 = 0\r\n" );
document.write( "\r\n" );
document.write( "Factor:\r\n" );
document.write( "\r\n" );
document.write( "(x + 1)(3x - 1) = 0\r\n" );
document.write( "\r\n" );
document.write( "x + 1 = 0;     3x - 1 = 0\r\n" );
document.write( "    x = -1;        3x = 1\r\n" );
document.write( "                    x = 1/3\r\n" );
document.write( "\r\n" );
document.write( "We put those points on a number line:\r\n" );
document.write( "\r\n" );
document.write( "------o-------o------\r\n" );
document.write( "     -1      1/3\r\n" );
document.write( "\r\n" );
document.write( "That determines three intervals, left of -1,\r\n" );
document.write( "between -1 and 1/3, and right of 1/3, which\r\n" );
document.write( "in interval notation are (-oo,-1), (-1,1/3), (1/3,oo).\r\n" );
document.write( "We do a first derivative test on each interval\r\n" );
document.write( "to determine whether the slope of the tangent line \r\n" );
document.write( "is positive or negative. It is increasing if the\r\n" );
document.write( "derivative, (slope of tangent line) is +, and\r\n" );
document.write( "decreasing if it is -.\r\n" );
document.write( "\r\n" );
document.write( "We make this chart to fill in, for each of the three\r\n" );
document.write( "intervals:\r\n" );
document.write( "\r\n" );
document.write( "Interval:     (-oo,-1) | (-1,1/3) | (1/3,oo) |\r\n" );
document.write( "Test pt.:              |          |          |\r\n" );
document.write( "Sign of f'             |          |          |\r\n" );
document.write( "incr or decr?          |          |          |\r\n" );
document.write( "\r\n" );
document.write( "For the test points, we choose any number on the\r\n" );
document.write( "interval, and we may as well choose the easiest\r\n" );
document.write( "one to substitute.  On the interval (-oo,-1), we\r\n" );
document.write( "will choose x=-2. On the interval (-1,1/3), we will\r\n" );
document.write( "choose x=0. On the interval (1/3,oo), we will\r\n" );
document.write( "choose x=1:\r\n" );
document.write( "\r\n" );
document.write( "Interval:     (-oo,-1) | (-1,1/3) | (1/3,oo) |\r\n" );
document.write( "Test pt.:        -2    |     0    |    1     |\r\n" );
document.write( "Sign of f'             |          |          |\r\n" );
document.write( "incr or decr?          |          |          | \r\n" );
document.write( "\r\n" );
document.write( "For the sign of f', we substitute each of those values\r\n" );
document.write( "in the equation for f'(x)\r\n" );
document.write( "\r\n" );
document.write( "f'(x) = 6x² + 4x - 2 \r\n" );
document.write( "f'(-2) = 6(-2)² + 4(-2) - 2\r\n" );
document.write( "f'(-2) = 6(4) - 8 - 2\r\n" );
document.write( "f'(-2) = 24 - 8 - 2\r\n" );
document.write( "f'(-2) =  14, which is + so we put a + for the sign of f'\r\n" );
document.write( "and since it is +, it is increasing, so we write \"incr\" \r\n" );
document.write( "on the bottom line.\r\n" );
document.write( "\r\n" );
document.write( "f'(x) = 6x² + 4x - 2 \r\n" );
document.write( "f'(0) = 6(0)² + 4(0) - 2\r\n" );
document.write( "f'(0) = 0 + 0 - 2\r\n" );
document.write( "f'(-2) =  -2, which is - so we put a - for the sign of f'\r\n" );
document.write( "and since it is -, it is decreasing, so we write \"decr\" \r\n" );
document.write( "on the bottom line.\r\n" );
document.write( "\r\n" );
document.write( "f'(x) = 6x² + 4x - 2 \r\n" );
document.write( "f'(1) = 6(1)² + 4(1) - 2\r\n" );
document.write( "f'(1) = 6(1) + 4 - 2\r\n" );
document.write( "f'(1) = 6 + 4 - 2\r\n" );
document.write( "f'(1) =  8, which is + so we put a + for the sign of f'\r\n" );
document.write( "and since it is +, it is increasing, so we write \"incr\" \r\n" );
document.write( "on the bottom line.  The chart looks like this: \r\n" );
document.write( "\r\n" );
document.write( "Interval:     (-oo,-1) | (-1,1/3) | (1/3,oo) |\r\n" );
document.write( "Test pt.:        -2    |     0    |    1     |\r\n" );
document.write( "Sign of f'        +    |     -    |    +     |\r\n" );
document.write( "incr or decr?   incr   |  decr    |  incr    | \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At -1, the point between the first two intervals, the\r\n" );
document.write( "graph changed from increasing to decreasing, so it\r\n" );
document.write( "reached a relative maximum point where x = -1,\r\n" );
document.write( "so the graph is increasing on the interval (-oo,-1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At 0, the point between the last two intervals, the\r\n" );
document.write( "graph changed from decreasing to increasing, so it\r\n" );
document.write( "reached a relative minimum point where x = 1/3,\r\n" );
document.write( "so the graph is decreasing on the interval (-1,1/3).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At 1/3, the point between the first two intervals, the\r\n" );
document.write( "graph changed from increasing to decreasing, so it\r\n" );
document.write( "reached a relative maximum point where x = 1/3,\r\n" );
document.write( "so the graph is increasing on the interval (1/3,oo).\r\n" );
document.write( "\r\n" );
document.write( "We check by drawing the graph:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The graph is going uphill to the right left of the\r\n" );
document.write( "point where x=-1, which is the relative maximum\r\n" );
document.write( "point (-1,4), so it is increasing on the interval\r\n" );
document.write( "left of that point.  \r\n" );
document.write( "\r\n" );
document.write( "Then the graph goes downhill to the right between\r\n" );
document.write( "the points where x=-1 and where x=1/3, which\r\n" );
document.write( "are the points (-1,4) and (1/3,44/27), which is\r\n" );
document.write( "a relative minimum point.  So the graph is decreasing\r\n" );
document.write( "on the interval between those two points. \r\n" );
document.write( "\r\n" );
document.write( "Then after leaving the point (1/3,44/27) the graph\r\n" );
document.write( "is going uphill to the right and thus is increasing \r\n" );
document.write( "right of that point.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );