The figure above shows the graph of $f'$, the derivative of a twice-differentiable function $f$, on the interval $[-3, 4]$. The graph of $f'$ has horizontal tangents at $x = -1$, $x = 1$, and $x = 3$. The areas of the regions bounded by the $x$-axis and the graph of $f'$ on the intervals $[-2, 1]$ and $[1, 4]$ are 9 and 12, respectively. (a) Find all $x$-coordinates at which $f$ has a relative maximum. Give a reason for your answer. (b) On what open intervals contained in $-3 < x < 4$ is the graph of $f$ both concave down and decreasing? Give a reason for your answer. (c) Find the $x$-coordinates of all points of inflection for the graph of $f$. Give a reason for your answer. (d) Given that $f(1) = 3$, write an expression for $f(x)$ that involves an integral. Find $f(4)$ and $f(-2)$.
The figure above shows the graph of $f'$, the derivative of a twice-differentiable function $f$, on the interval $[-3, 4]$. The graph of $f'$ has horizontal tangents at $x = -1$, $x = 1$, and $x = 3$. The areas of the regions bounded by the $x$-axis and the graph of $f'$ on the intervals $[-2, 1]$ and $[1, 4]$ are 9 and 12, respectively.
(a) Find all $x$-coordinates at which $f$ has a relative maximum. Give a reason for your answer.
(b) On what open intervals contained in $-3 < x < 4$ is the graph of $f$ both concave down and decreasing? Give a reason for your answer.
(c) Find the $x$-coordinates of all points of inflection for the graph of $f$. Give a reason for your answer.
(d) Given that $f(1) = 3$, write an expression for $f(x)$ that involves an integral. Find $f(4)$ and $f(-2)$.