Chapter 3 - Applications of Differentiation - 3.3 How Derivatives Affect the Shape of a Graph - 3.3 Exercises - Page 228: 7

a) $x = 3; x=5$ b) $x = 2; x=4; x = 6$ c) $x = 1; x = 7$

a) There is an IP at $x = 3$ because the graph of $f$ changes from CD to CU there. There is an IP at $x = 5$ because the graph of $f$ changes from CU to CD there. b) There is an IP at $x = 2$ and at $x = 6$ because $f'(x)$ has a maximum value there, and so $f''(x)$ changes from positive to negative there. There is an IP at $x = 4$ because $f'(x)$ has a minimum value there and so $f''(x)$ changes from negative to positive there. c) There is an inflection point at $x = 1$ because $f''(x)$ changes from negative to positive there, and so the graph of $f$ changes from concave downward to concave upward. There is an inflection point at $x = 7$ because $f''(x)$ changes from positive to negative there, and so the graph of $f$ changes from concave upward to concave downward.