Chapter 2 - Section 2.8 - The Derivative as a Function - 2.8 Exercises - Page 162: 3

Function b - derivative IV Function a - derivative II Function d - derivative III Function c - derivative I

b) In the graph of function $b$, we see that there are 2 pointy points. There can't be any tangent lines at that point, which means that the graph of its derivation cannot be continuous. In other words, we would look for a graph of the derivative where there are 2 points which are not defined. Graph IV fits this description. (a) The graph of function $a$ has 2 points where the graph changes direction from going up to going down and vice versa. At that point, the tangent line is parallel to the $Ox$, which means the slope of the tangent is $0$. In other words, the value of the derivative there is $0$. Therefore, we would look for a graph of derivative which has 2 points where $y=0$. Graph II fits this description. (d) The same argument for (a) is also true for (d). There are 3 points where graph $d$ changes direction from up to down, or down to up. That means its graph of derivative must have 3 points where $y=0$. Graph III fits this description. (c) We can use the same argument for (c). There is only one time it changes from going down to up, which means the graph of derivative has 1 point where $y=0$. Graph I has such a thing.