0 Question 1 v Given the function g(x) = 43:3 + 12.782 963:, find the first derivative, g'(a:). g'lw) =l J Notice that g'(a:) = 0 when a: = 4, that is, g'(4) = 0. Now, we want to know whether there is a local minimum or Local maximum at a: = 4, so we will use the second derivative test. Find the second derivative, g"(:c). 9"(=v) =i ' Evaluate g"(4). g"(4) = Based on the sign of this number, does this mean the graph of g(a:) is concave up; or concave down at a: = 4? [Answer either up or down -- watch your spelling!!] At a: = 4 the graph of 9(a)) is concave Based on the concavity of 9(3)) at a: = 4, does this mean that there is a local minimum or local maximum at a: = 4? [Answer either minimum or maximum -- watch your spelling!!] At a: = 4 there is a local Check Answer 0 Question 2 v The function f(:c) = 2:133 + 33:1:2 144m + 11 has one local minimum and one local maximum. This function has a local minimum at a: =' ' with value ' ' and a local maximum at :c = ' ' with value ' ' Check Answer 0 Question 3 v Consider the function ns) 2 4:1: + Elm1. For this function there are four important intervals: (00, A], [A, B),(B, C], and [0, 00) where A, and C are the critical numbers and the function is not defined at B. Find A l and B i i and C i ' For each of the following open intervals, tell whether m) is increasing or decreasing. (oo,A)= (AB): (520): (0,00) Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(m) is concave up or concave down. (00,B)= (300): Check Answer 0 Question 4 v Consider the function f(:1:) = $2653. f(:1:) has two inflection points at x = C and x = D with C Consider the function f(:1:) = 123:5 l 453:4 360:1:3 + 1. x) has inflection points at (reading from left to right) x = D, E, and F where D is ' ' and E is ' \ ' and F is l ' For each of the following intervals, tell whether f(m) is concave up or concave down. (-00,D)= (DJ-*7): (Em): (F, co): Check