19 Goal is to sketch the curve for f(x) defined as follows { f(x) = [ (In x) / ( sqrt(x))] [NOTE: the implied domain is (x > 0} 1 a) Find the first derivative f(x), and use this to determine all critical points. b) Find regions where increasing and regions where decreasing. Use this to classify each critical point as a relative maximum, relative minimum, or neither. c) Find the second derivative f'(x) and determine where this is zero. Use this to classify regions of concavity and identify any inflection points. [Note the limit at infinity is zero, leading to a horizontal asymptote of y=0 along the positive x-axis. ] d) Sketch the curve QUESTION 20 Optimization A corral with three separate regions is constructed against a river as indicated in the diagram below. The amount of fence material available is xyz. Find the dimensions that will maximize the total area enclosed (and also this total area) { river Give the objective and constraint equations. Reduce the objective to one variable and use the derivative to maximize. QUESTION 22 antiderivative problem Consider a moving particle whose acceleration is given by the following function { a(t) = t- 3+ sin t } a) Find the velocity function v(t) given that the initial velocity is v_0 = 5. b) Find the position function r(t) given that the initial position is r_0 = 10.