Question

3c and 3d

3. Given the function defined by fx)-2x3x-12x+ 20. a) By using Rational Root Theorem and Synthetic Division, write flk) as a product of its factors dentily the zeros. b) The slope function can be found by using the formula: (lnxWrite the equation of the line tangent to the graph offat x 0. c) A tangent line drawn to the graph of f is said to be a horizontal tangent or parallel to x-axis if the slope is zero. Find the x-and y-coordinates of all points on the graph of f where the line tangent to the graph is parallel to the x-axis. d) The points that you found in part c are called the extrema. Sketch manually the graph of f(x) by using the coordinates of these points, y-intercept, the zeros that you found in part a by using an appropriate scale. Then classify these extrema as relative or absolute maximum or minimum.

Answer 1

"F(X) = 2X^3 - 3X^2 - 12 X + 20 F^1(X) = 6X^2 - 6X - 12 state of tangent line of curve y = F(x) is given by F(X) then slope of tangent line = F^1 = 6X^2 - 6X - 12 Now slope of tangent line = 0 implies F^1(X) = 0 implies 6X^2 - 6X - 12 = 0 implies 6(X^2 - X - 2) = 0 implies X^2 - X - 2 = 0 implies X^2 - 2X + X - 2 = 0 X(X - 2) + 1(X - 2) = 0 implies (X - 2)(X + 2) = 0 implies X = 2, - 1 F""(X) = 12 times - 6 f""(2) = 12 times 2 - 6 = 18 > 0 (min) F""(- 1) = -12 - 6 = -18 < 0(max) hence at X = - 1 maximum occurs and at x = 2 minimum occurs "