Answer
a) $x=-2,-1,1,3$
b) $x \in (-infty, -2) \cup (-1,1) \cup (3, +\infty)$
c) $x \in (-2, -1) \cup (1,3)$
d) $g''(0)=4$
Work Step by Step
a) $y$ has horizontal tangent lines when $f'(x) = 0$. From the graph, this occurs at $x=-2,-1,1,3$.
b) $y$ has tangent lines with positive slope when $f'(x)>0$. From the graph, this occurs at $x \in (-\infty, -2) \cup (-1,1) \cup (3, +\infty)$.
c) $y$ has tangent lines with negative slope when $f'(x)<0$. From the graph, this occurs at $x \in (-2, -1) \cup (1,3)$.
d) From the graph, $f'(0) = 2$ and $f''(0) = 0$ as the $f'(x)$ has a relative maximum at $x=0$.
Then, given $g(x) = f(x)sin(x)$:
$g'(x) = f'(x)sin(x) + f(x)cos(x)$
$g''(x) = f''(x)sin(x) + f'(x)cos(x) + f'(x)cos(x) -f(x)sin(x)$
$g''(0) = f''(0)sin(0) + f'(0)cos(0) + f'(0)cos(0) -f(0)sin(0) = 0*0 + 2 * 1 + 2* 1 - 0*0 = 4$
