Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 3

Below are the graphs of functions and its derivatives. Match each function with the graph of its
derivatives and give reasons to your choices.









Graph(a) matches Figure II because graph(a) is symmetric to the origin so the graph of its derivative
is symmetric to the $y$-axis. Also, graph(a) has a maximum positive slope at $x=0$. So the graph of its
derivative has a maximum value at $x=0$.



Graph(b) matches Figure IV because graph(b) is symmetric to the origin, so the graph of its derivative is symmetric
to the $y$-axis. Also, graph(b) have points where the function is not differentiable so the graph of its derivative
has discontinuity.



Graph(c) matches Figure I because graph(c) is symmetric about the $y$-axis, so the graph of its derivative is symmetric
to the origin. Also, graph(c) has a horizontal tangent at $x=0$ that makes the derivative zero at $x=0$. Lastly the
derivative is positive when graph(c) is increasing and the derivative is negative when graph(c) is decreasing.



Graph(d) matches Figure III because graph(d) is symmetric about the $y$-axis, so the graph of its derivative is symmetric
to the origin. Also, graph(d) has horizontal tangent at $x=0$ that makes the derivative 0 at $x=0$. Lastly, the function has
three critical points where its concavity changes, thus the graph of the derivatives has three zeros.