Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above. (a) On what intervals, if any, is $f$ increasing? Justify your answer. (b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer. (c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$. (d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.
Let $f$ be a function defined on the closed interval $-3 \leq x \leq 4$ with $f(0) = 3$. The graph of $f'$, the derivative of $f$, consists of one line segment and a semicircle, as shown above.\\
(a) On what intervals, if any, is $f$ increasing? Justify your answer.\\
(b) Find the $x$-coordinate of each point of inflection of the graph of $f$ on the open interval $-3 < x < 4$. Justify your answer.\\
(c) Find an equation for the line tangent to the graph of $f$ at the point $(0, 3)$.\\
(d) Find $f(-3)$ and $f(4)$. Show the work that leads to your answers.