# Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 302: 36

(a) $f$ is increasing on these intervals: $(1,6)\cup (8,\infty)$ $f$ is decreasing on these intervals: $(0,1)\cup (6,8)$ (b) $f$ has a local maximum at $x=6$ $f$ has a local minimum at $x=1$ and $x=8$ (c) $f$ is concave up on these intervals: $(0,2)\cup (3,5)\cup (7,\infty)$ $f$ is concave down on this interval: $(2,3)\cup (5,7)$ (d) The x-coordinates of the points of inflection are $x=2, x=3, x=5,$ and $x=7$ (e) We can see a sketch of the graph of $f$ below. (a) $f$ is increasing when $f'(x) \gt 0$ $f$ is increasing on these intervals: $(1,6)\cup (8,\infty)$ $f$ is decreasing when $f'(x) \lt 0$ $f$ is decreasing on these intervals: $(0,1)\cup (6,8)$ (b) $f$ has a local maximum when $f'(x)$ changes from positive to negative. $f$ has a local maximum at $x=6$ $f$ has a local minimum when $f'(x)$ changes from negative to positive. $f$ has a local minimum at $x=1$ and $x=8$ (c) $f$ is concave up when $f'(x)$ is increasing. $f$ is concave up on these intervals: $(0,2)\cup (3,5)\cup (7,\infty)$ $f$ is concave down when $f'(x)$ is decreasing. $f$ is concave down on this interval: $(2,3)\cup (5,7)$ (d) A point of inflection is a point where the graph $f$ changes concavity. $f$ changes concavity at $x = 2, 3, 5, 7$ so the x-coordinates of the points of inflection are $x=2, x=3, x=5,$ and $x=7$ (e) We can see a sketch of the graph of $f$ below. 