It is given that $f(x)=2\sin\frac{x}{2}$ and $g(x)=3\cos x+1$, where $0 \le x \le 2\pi$.

1. Identify the coefficient of $x$ and calculate the period

For $f(x) = 2\sin\left(\frac{1}{2}x\right)$, the coefficient of $x$ is $\frac{1}{2}$.
$\text{Period} = \frac{2\pi}{1/2} = 4\pi$

1. Determine the smallest value of the function in the given domain

For $0 \le x \le 2\pi$, the angle $\frac{x}{2}$ ranges from $0$ to $\pi$.
In this interval, $0 \le \sin\frac{x}{2} \le 1$.
Therefore, the smallest value of $2\sin\frac{x}{2}$ is $2(0) = 0$.

1. Determine the maximum value of the cosine function

The maximum value of $\cos x$ is $1$.

2. Calculate the largest value of $g(x)$

Largest value of $g(x) = 3(1) + 1 = 4$

1. Determine the key features of $y=f(x)=2\sin\frac{x}{2}$ for $0 \le x \le 2\pi$

Amplitude $= 2$. Period $= \frac{2\pi}{1/2} = 4\pi$.
For $0 \le x \le 2\pi$, the graph is half of a sine wave.
Key points: $(0,0)$, maximum at $(\pi, 2)$, and $(2\pi, 0)$.

2. Determine the key features of $y=g(x)=3\cos x+1$ for $0 \le x \le 2\pi$

Amplitude $= 3$. Period $= 2\pi$. Vertical shift $= +1$.
Maximum value $= 3+1=4$, minimum value $= -3+1=-2$.
Key points: $(0,4)$, $(\frac{\pi}{2}, 1)$, $(\pi, -2)$, $(\frac{3\pi}{2}, 1)$, $(2\pi, 4)$.

3. Sketch both graphs on the same axes and label them

Draw the axes. Plot the key points for $f(x)$ and draw a smooth curve. Plot the key points for $g(x)$ and draw a smooth curve. Label the curves $y=f(x)$ and $y=g(x)$.

1. Interpret the inequality $f(x) > g(x)$ graphically

The inequality $f(x) > g(x)$ corresponds to the region where the graph of $y = f(x)$ lies above the graph of $y = g(x)$.

2. Determine the range of $x$ using the intersection points

From the sketch in part (d), the curve $y = 2\sin\frac{x}{2}$ starts at $(0,0)$, reaches a maximum of $2$ at $x=\pi$, and ends at $(2\pi, 0)$. The curve $y = 3\cos x + 1$ starts at $(0,4)$, goes down to a minimum of $-2$ at $x=\pi$, and goes back up to $(2\pi, 4)$.
The graphs intersect at $x=a$ and $x=b$ (where $a < b$). Between these two points, the sine curve $f(x)$ is above the cosine curve $g(x)$.
Therefore, $f(x) > g(x)$ for $a < x < b$.