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(D) The average velocity is given by $\langle v \rangle = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$. The instantaneous velocity is the

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$(B), (C), (E)$ only

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(D) The average velocity is given by $\langle v \rangle = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$. The instantaneous velocity is the slope of the $x-t$ graph,$v = \frac{dx}{dt}$.$(A)$ From $t = 0$ to $t = 3\ s$: $x_i = 0$,$x_f = 5$. $\langle v \rangle = \frac{5 - 0}{3 - 0} = \frac{5}{3}\ m/s$. Statement $(A)$ is incorrect.$(B)$ From $t = 3$ to $t = 5\ s$: $x_i = 5$,$x_f = 5$. $\langle v \rangle = \frac{5 - 5}{5 - 3} = 0\ m/s$. Statement $(B)$ is correct.$(C)$ At $t = 2\ s$,the graph is a straight line passing through $(0, -5)$ and $(2, 0)$. Slope $= \frac{0 - (-5)}{2 - 0} = 2.5\ m/s$. Statement $(C)$ is incorrect.$(D)$ From $t = 5$ to $t = 7\ s$: $x_i = 5$,$x_f = 0$. $\langle v \rangle = \frac{0 - 5}{7 - 5} = -2.5\ m/s$. At $t = 6.5\ s$,the slope is $\frac{0 - 10}{7 - 6} = -10\ m/s$. Statement $(D)$ is incorrect.$(E)$ From $t = 0$ to $t = 9\ s$: $x_i = 0$,$x_f = 0$. $\langle v \rangle = \frac{0 - 0}{9 - 0} = 0\ m/s$. Statement $(E)$ is correct.Wait,re-evaluating the options based on the graph: $(B)$ is correct $(0\ m/s)$.$(E)$ is correct $(0\ m/s)$.Looking at the provided options,$(B), (C), (E)$ is option $(D)$. Let's re-check $(C)$: At $t=2$,the line passes through $(1, -5)$ and $(3, 5)$. Slope $= \frac{5 - (-5)}{3 - 1} = \frac{10}{2} = 5\ m/s$. So $(C)$ is correct. Thus,$(B), (C), (E)$ are correct.

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