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# Class Note for MATH 125 with Professor Rychlik at UA

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15
A cubic function Without a critical point BY MAREK RYCHLIK Lecture of November 3 2008 Example 1 Let us consider the function yzg12312 Find its critical points in ection points and intervals of monotonicity Determine how many roots the function will have Solution First we nd the critical points by solving the equation y 3122130 This is a quadratic equation A refresher from algebra the equation a12bx c0 has 2 1 or zero solutions depending on the sign of the discriminant A b2 7 4 a c If A gt 0 then there are two roots given by formulas 11 2a 7 b 7 VA 12 7 2a For our case A 22 7 4 3 3 7 32 Hence the equation has no roots Thus there are no crit ical points Moreover y gt 0 because the coe icient at x2 is positive The graph of y which is a parabola lies above the z axis The in ection point is determined from the equation y 6z2 0 which gives 13 7 1 In summary the function is strictly increasing there are no critical points and there is one in ection point at z 7 The in ection point is at the minimum point of the rst derivative Thus the function changes most slowly at the in ection point There will be exactly one root This is deduced from the following facts a The limit of the function y fx 1312 3x 2 is ioo as 1 goes to ioo Thus the function is positive for large positive I and negative for large negative 1 Because the function is continuous it must cross the z axis See for example the Racetrack Principle of Section 910 which is a special version of the Intermediate Value Theorem see Wikipedia b A strictly increasing function such as fx may intersect the z axis at most once lnceed if 1 lt I then fzl lt fzg which means that fzl u and thus only one of the numbers fzl fzg can be zero Below is a graph obtained with a free program called GNUplot GNUplot plot X 11 fxx3x23x2 fx fx 1 w x 1 05 0 05 GNUplot

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