$\begin{array}{l}\mathrm{x}\left(\mathrm{t}\right) = {\mathrm{x}}_0(1 - {\mathrm{e}}^{-\mathrm{\gamma t}})\end{array}$

$\mathrm{When} \mathrm{t} = 0; \mathrm{x}\left(\mathrm{t}\right) = {\mathrm{x}}_0(1-{\mathrm{e}}^{-0}) = {\mathrm{x}}_0(1-1) = 0.$

Step 2: Calculate the velocity, v(t) = $\frac{\mathrm{dx}}{\mathrm{dt}}$

$\mathrm{v}\left(\mathrm{t}\right) = \frac{\mathrm{dx}\left(\mathrm{t}\right)}{\mathrm{dt}} = {\mathrm{x}}_0{\mathrm{\gamma e}}^{-\mathrm{\gamma t}}.$

Step 3: Calculate initial velocity by putting t = 0

V(0) = ${\mathrm{x}}_0{\mathrm{\gamma e}}^0 = {\mathrm{x}}_0\mathrm{\gamma }.$

(b)Hint: If f is increasing $\frac{\mathrm{df}}{\mathrm{dt}}$ > 0, if decreasing $\frac{\mathrm{df}}{\mathrm{dt}}$ < 0.

Step 1: Check weather f is increasing or decreasing

For x(t)

As here $\frac{\mathrm{dx}}{\mathrm{dt}}$ = ${\mathrm{x}}_0{\mathrm{\gamma e}}^{-\mathrm{\gamma t}}$ > 0 

i.e., x is increasing

For velocity, v(t)

$\frac{dv\left(t\right)}{dt} = -x_0{\gamma }^2{e}^{-\gamma t} < 0$

i.e., v(t) is decreasing 

For acceleration, a(t)

$\frac{da\left(t\right)}{dt} = x_0{\gamma }^3{e}^{-\gamma t} >0$

i.e., a(t) is increasing

Step 2: Calculate maximum and minimum values

For increasing, the maximum value is at t = $\infty$ and minimum value is at t = 0,

For decreasing, the maximum value is at t = 0 and the minimum value is at t = $\infty$.

For x(t)

As it is increasing

So,

$\begin{array}{ll}\mathrm{x}\left(\mathrm{t}\right)\text{ is maximum when }\mathrm{t}=\mathrm{\infty }; \mathrm{x}\left(\mathrm{t}\right){]}_{\max }={\mathrm{x}}_0& \\ \mathrm{x}\left(\mathrm{t}\right)\text{ is minimum when }\mathrm{t}=0; \mathrm{x}\left(\mathrm{t}\right){]}_{\min }=0& \\ v\left(t\right) is decrea\sin g& \\ So,& \\ \mathrm{v}\left(\mathrm{t}\right)\text{ is maximum when }\mathrm{t}=0; \mathrm{v}\left(0\right)={\mathrm{x}}_0\mathrm{\gamma }& \\ \mathrm{v}\left(\mathrm{t}\right)\text{ is minimum when }\mathrm{t}=\mathrm{\infty }; \mathrm{v}\left(\mathrm{\infty }\right)=0& \\ a\left(t\right) is increa\sin g & \\ So,& \\ \mathrm{a}\left(\mathrm{t}\right)\text{ is maximum when }\mathrm{t}=\mathrm{\infty }; \mathrm{a}\left(\mathrm{\infty }\right)=0& \\ \mathrm{a}\left(\mathrm{t}\right)\text{ is minimum when }\mathrm{t}=0; \mathrm{a}\left(0\right)=-{\mathrm{x}}_0{\mathrm{\gamma }}^2& \end{array}$