Let $f$ be a differentiable function with $f(4) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f'$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above. (a) Find $f(0)$ and $f(5)$. (b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer. (c) Let $g$ be the function defined by $g(x) = f(x) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$? Show the analysis that leads to your answer. (d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.
Let $f$ be a differentiable function with $f(4) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f'$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.
(a) Find $f(0)$ and $f(5)$.
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.
(c) Let $g$ be the function defined by $g(x) = f(x) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$? Show the analysis that leads to your answer.
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.