The function f is defined by $f(x)=4\cos ax + b$ for $0 \le x \le 3\pi$, where $a$ and $b$ are constants. The maximum value of f is 6 and the period of f is $4\pi$.

1. State the amplitude from the given function

The amplitude of $f(x) = 4\cos ax + b$ is given by the coefficient of the cosine term.
$\text{Amplitude} = 4$

1. Find the value of $a$ using the given period.

The period of $f(x) = 4\cos ax + b$ is given by $\frac{2\pi}{a}$.
$\frac{2\pi}{a} = 4\pi$
$a = \frac{2\pi}{4\pi} = \frac{1}{2}$

2. Find the value of $b$ using the maximum value and amplitude.

The maximum value of $f(x)$ is the amplitude plus the vertical shift $b$.
$4 + b = 6$
$b = 2$

1. Identify the full equation of the function from earlier parts

From parts (i) and (ii), the amplitude is $4$, the period is $4\pi$, so $a = \frac{2\pi}{4\pi} = \frac{1}{2}$.
The maximum value is $6$, so $4 + b = 6 \implies b = 2$.
Thus, $f(x) = 4\cos\left(\frac{1}{2}x\right) + 2$.

2. Calculate key coordinates for the sketch within the given domain

Evaluate $f(x)$ at key intervals of the period ($0$, $\pi$, $2\pi$, $3\pi$):

• At $x = 0$, $y = 4\cos(0) + 2 = 6$
• At $x = \pi$, $y = 4\cos\left(\frac{\pi}{2}\right) + 2 = 2$
• At $x = 2\pi$, $y = 4\cos(\pi) + 2 = -2$
• At $x = 3\pi$, $y = 4\cos\left(\frac{3\pi}{2}\right) + 2 = 2$

3. Find the x-intercept(s)

Set $f(x) = 0$:
$4\cos\left(\frac{1}{2}x\right) + 2 = 0$
$\cos\left(\frac{1}{2}x\right) = -\frac{1}{2}$
$\frac{1}{2}x = \frac{2\pi}{3} \implies x = \frac{4\pi}{3}$

4. Sketch the graph

Draw a smooth curve passing through $(0, 6)$, $(\pi, 2)$, $\left(\frac{4\pi}{3}, 0\right)$, $(2\pi, -2)$, and ending at $(3\pi, 2)$.