Answer
(a) Using the graph of $f(x)$, we can make the following estimates:
The graph is concave down in these intervals: $(0,1.1)\cup (1.6, 2.1)$
The graph is concave up in these intervals: $(1.1, 1.6)\cup (2.1, 3.1)$
The points of inflection are $(1.1, 0), (1.6, 0),$ and $(2.1, 0)$
(b) Using the graph of $f''(x)$, we can make the following estimates:
The graph is concave down in these intervals: $(0,0.85)\cup (1.57, 2.29)$
The graph is concave up in these intervals: $(0.85, 1.57)\cup (2.29, 3.14)$
The points of inflection are $(0.85, 0.74), (1.57, 0),$ and $(2.29, -0.73)$
Work Step by Step
(a) $f(x) = sin~2x+sin~4x,~~~~0 \leq x \leq \pi$
Using the graph of $f(x)$, we can make the following estimates:
The graph is concave down in these intervals: $(0,1.1)\cup (1.6, 2.1)$
The graph is concave up in these intervals: $(1.1, 1.6)\cup (2.1, 3.1)$
The points of inflection are $(1.1, 0), (1.6, 0),$ and $(2.1, 0)$
(b) $f'(x) = 2~cos~2x+4~cos~4x$
$f''(x) = -4~sin~2x-16~sin~4x$
Using the graph of $f''(x)$, we can make the following estimates:
The graph is concave down in these intervals: $(0,0.85)\cup (1.57, 2.29)$
The graph is concave up in these intervals: $(0.85, 1.57)\cup (2.29, 3.14)$
We can find the points of inflection:
$f(0.85) = sin~2(0.85)+sin~4(0.85) = 0.74$
$f(1.57) = sin~2(1.57)+sin~4(1.57) = 0$
$f(2.29) = sin~2(2.29)+sin~4(2.29) = -0.73$
The points of inflection are $(0.85, 0.74), (1.57, 0),$ and $(2.29, -0.73)$
