Step 1: Understand areal velocity. As a planet orbits the sun, the line joining them sweeps out area. Areal velocity is how much area is swept per second. By Kepler's second law this value stays constant throughout the orbit.
Step 2: Write the areal velocity formula. The area swept per unit time is: \[ \frac{dA}{dt} = \frac{1}{2} r v \] Here $r$ is the distance from the sun and $v$ is the speed of the planet at that point.
Step 3: Think about where speed is smallest. Because the product $r v$ is fixed, the speed is smallest when $r$ is largest. The largest distance is given, so this is where the minimum speed occurs.
Step 4: Rearrange for the speed. Solve the formula for $v$: \[ v = \frac{2}{r} \cdot \frac{dA}{dt} \] At the maximum distance this gives the minimum speed.
Step 5: Put in the numbers. The areal velocity is $4 \times 10^{16}\text{ m}^2\text{s}^{-1}$ and the maximum distance is $4 \times 10^{12}\text{ m}$: \[ v_{min} = \frac{2 \times 4 \times 10^{16}}{4 \times 10^{12}} \]
Step 6: Simplify the result. The top is $8 \times 10^{16}$ and dividing by $4 \times 10^{12}$ gives: \[ v_{min} = 2 \times 10^{4}\text{ ms}^{-1} \] So the minimum speed is two times ten to the fourth metres per second. \[ \boxed{2 \times 10^{4}\text{ ms}^{-1}} \]