The function $f$ is defined on the closed interval $[-2, 8]$ and satisfies $f(2) = 1$. The graph of $f'$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure. (a) Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer. (b) On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer. (c) Find the value of $\lim_{x \rightarrow 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, or show that it does not exist. Justify your answer. (d) Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. Justify your answer.
The function $f$ is defined on the closed interval $[-2, 8]$ and satisfies $f(2) = 1$. The graph of $f'$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure.
(a) Does $f$ have a relative minimum, a relative maximum, or neither at $x = 6$? Give a reason for your answer.
(b) On what open intervals, if any, is the graph of $f$ concave down? Give a reason for your answer.
(c) Find the value of $\lim_{x \rightarrow 2} \frac{6f(x) - 3x}{x^{2} - 5x + 6}$, or show that it does not exist. Justify your answer.
(d) Find the absolute minimum value of $f$ on the closed interval $[-2, 8]$. Justify your answer.