Look over the following two schematic conditions:
simulate this circuit – Schematic created using CircuitLab
Your value of \$R_i\$ must be small enough to support a load current of \$100\:\textrm{mA}\$ (which implies a load of \$\tfrac{12\:\textrm{V}}{100\:\textrm{mA}}=120\:\Omega\$.) You also have to support the minimum needed current through \$D_1\$ at this time, as well. For a typical zener like this, that value can span over some range. But for good regulation, you probably want \$I_z\ge 10\:\textrm{mA}\$. And \$R_i\$ must do all of this while the voltage supply is at its minimum value of \$15\:\textrm{V}\$. So:
$$R_i\le\frac{15\:\textrm{V}-12\:\textrm{V}}{100\:\textrm{mA}+10\:\textrm{mA}}\approx 27.27\: \Omega$$
A standard value of \$27\:\Omega\$ would do.
Note that I had to place this circuit at the extremes in order to make sure that \$R_i\$ was having to cope with the worst possible supply circumstance (minimum) it might experience when trying to deal with the worst case load circumstance (maximum.)
Now that we know that \$R_i=27\:\Omega\$, the second schematic allows us to change things up in order to compute the worst case power in \$R_i\$ (and in \$D_1\$.) The power in \$R_i\$ will be the same, regardless of the load current. But I'm setting the load current to a minimum now, while setting the supply to its maximum, in order to see the worst case for \$D_1\$ (whose power does depend on the load current.)
In this case, the current through \$R_i\$ is:
$$I_{R_i}=\frac{20\:\textrm{V}-12\:\textrm{V}}{27\:\Omega}\approx 300\:\textrm{mA}\approx 2.5\:\textrm{W}$$
The power in \$R_i\$ is then about \$27\:\Omega\cdot\left(300\:\textrm{mA}\right)^2\$
Since only \$20\:\textrm{mA}\$ of that is going through the load, the remainder, or \$280\:\textrm{mA}\$, is left to go through \$D_1\$. So the power required by the zener diode is \$12\:\textrm{V}\cdot 280\:\textrm{mA}\approx 3.4\:\textrm{W}\$.
That covers (a) and (b) pretty well.
Now the problem is about (c). Before I engage that issue, I want to say that the datasheet for the 1N4742A says that when using \$21\:\textrm{mA}\$, \$R_z=9\:\Omega\$. This would tend to mean that if someone is telling me that \$R_z=3\:\Omega\$, the value I used for the minimum \$I_z=10\:\textrm{mA}\$ wasn't enough. It should be much higher. And this would tend to mean that I didn't compute \$R_i\$ correctly. However, your problem doesn't require this particular zener and so I can't debate or argue the facts, as given. The problem also didn't state a minimum zener current to use. So this allows me to keep my calculations and just move on.
Since the dynamic resistance of the zener is given as \$R_z=3\:\Omega\$, and since I can see that the dynamic change in \$I_z\$, from minimum to maximum, is \$300\:\textrm{mA}-10\:\textrm{mA}=290\:\textrm{mA}\$, I can multiply these two to get \$870\:\textrm{mV}\$ variation at the zener. Or, if I'm lucky with the zener, I may get \$12\:\textrm{V}\pm 440\:\textrm{mV}\$ at the output over all design circumstances (excepting thermal drift and aging and part variation, which also don't seem to be part of the problem here, but would be in the case of a real circuit.)
But let's do this a different way. We can insert \$R_z\$ into the circuit:
simulate this circuit
Now, that's not an exact reality. I just picked an arbitrary \$11.55\:\textrm{V}\$ for the ideal zener value in order to make things work out to center on \$12\:\textrm{V}\$ at the output. The reality will be different. Worse, the dynamic range of current through the zener is a factor of 30! There is no possible way that \$R_z=3\:\Omega\$ for such a dynamic range of current. It might be valid had I chosen a minimum \$I_z\ge 100\:\textrm{mA}\$, perhaps. Which would have resulted in a different value for \$R_i\$. But the problem doesn't care, so I don't care. Had it listed a specific part where I could examine a datasheet, a more accurate idea might have developed from here.
But the above works for homework, I suppose.
Best Answer
5mA 12.2V
1mA 12.0V
100uA 11.7V
10uA 11.4V
1uA 11.1V
100nA 10.8V
10nA 10.5V
1nA 10.1V
Edit: To address the particular problem as described somewhat in the comments:
To design a shunt regulator with a resistor and a zener (or TL431 or similar shunt regulator).
The series resistor has to supply all the load current, plus a small amount to keep the regulator working, at the minimum input voltage. Suppose you have an output voltage of 12V and a maximum load current of 100mA and a minimum load current of 0mA. Input voltage is 16V maximum and 14V minimum.
The resistor has to pass 100mA with 2V across it plus whatever the regulator needs. Let's assume that is zero. So the resistor has to be no higher than 20 ohms. Given say 5% tolerance and E24 values, maybe we'd use 18 ohms. Now we have at least 100mA for the load and minimum 5.8mA for the regulator. So we don't need to worry about minimum regulator current unless it's less than 6mA, just the tolerance takes care of it.
If the load now goes to 0mA the regulator has to eat all that current (106mA) so it dissipates 1.3W. That's a lot of power. But it gets worse.
Now, suppose we have 16V input, the resistor current is now (16-12)/18 = 222mA nominal. With our load drawing 100mA the shunt regulator has to eat 122mA, causing it to dissipate more than 1.4W. And if the load current goes to zero, then it has to eat the entire current, and dissipate about 2.7W of power, which would require a large device with a large heatsink.
You can see why series regulators are preferred for high load currents, and when the input voltage ranges above but also gets close to the output voltage.
I didn't address the quality of the regulation here, but the differential resistance of the zener you mentioned is similar to the series resistor we calculated at minimum input voltage. That means that (at minimum input voltage) a 0.1V change in input voltage will cause about 0.05V in output voltage, which is not very good regulation.