4.1 Increasing and Decreasing Functions 1. Find the intervals of increase and decrease for the function represented graphically to the right. 2 2. Find the intervals of increase and decrease. a) f (x ) = _ b ) f ( x ) = Vx 3. Find the intervals of increase and decrease for f(x) =4x3+ 15x2 -18x+1. 4. Find the intervals of increase and decrease for f (x) = Vx2 -4 . 5. What conditions should the coefficients a , b , c, and d satisfy in order that the function f(x) = ax + bx2 + cx + d is decreasing at any number.4.2 Critical Points. Local Maxima and Minima 1. Find the local (relative) extremum (minimum or maximum) points for the function represented to the right. 2. Find the critical point(s) for the function represented to the right. . 3. For each case, find the critical points. a) f(x) = x3 3x b) f(x) = x2 1 4. Find the local (relative) extrema for f(x) = 3x4 + 4x3 2 . 5. Find a cubic function of the form f (x) = mix3 +iisix2 + cx+d with local extremum points at (0,4) and (2,0). 4.3 Vertical and Horizontal Asymptotes 1. Find the equation of any asymptote for the function represented graphically to the right. -3 2. For each case, find the equation of any asymptote. a ) f (x ) = ( x+1)2 x-1 x2 - 1 b) f (x ) = X x2 + 3x+ 2 c) f (x) = Vx - 1 3. Find the equation of the oblique asymptote (if exists). a) f ( x ) = - 2x- +1 b) f (x ) = x(x - 1) 2 x-1 x-+14.4 Concavi and Points of lnflection 1. Find the intervals on which the graph, given to the right, is , concave upward or downward. Then identify the point(s) of y I EIINIDI inflection. IIIUWI.' 2. For each case, find the intervals of concavity. 3) f(x) =x2 +3 b) f(x) =2x3 + 3.1:2 c) f(x) = xl 3. Find the point(s) of inflection for f(x) = x4 2x.3 + x 1. 4. Find a cubic function of the form f (x) =ax3 + in:2 + at + d with a point of inflection at (0,0) and a local extremum point at (2,2). 5. Find a polynomial function of degree four with two inflection points at (0,0) and (2,6) and a zero at (1,0) . 4.5 An Algorithm for Curve Sketching 1. Sketch the graph of the quartic function y = f (x) = (x2 -4)(x2 -2) . 2. Sketch the graph of the rational function y = f (x) = x-+1 3. Sketch the graph of the rational function y = f (x) = - x2 - 1