zener diode circuit analysis
Indeed the explanation of a) looks weird, but the issue is that \$V_Z\$ (= 15 V) is defined at \$I_{ZT}\$ (= 17 mA) and not at the minimum zener current \$I_{ZK}\$ (= 0.25 mA).

So we start with 15 V (which is for \$I_{ZT}\$ = 17 mA), determine that the zener diode's internal resistance \$R_Z\$ drops 14 ohm * 17 mA = 0.238 V Which results in 15 V - 0.238 V = 14.762 V across the "internal" Zener diode (without the drop of \$R_Z\$).

As the maximum value of \$R\$ is asked for, we want the smallest current to flow, that smallest current is \$I_{ZK}\$ = 0.25 mA. At this 0.25 mA the drop across \$R_Z\$ will be 14 ohms * 0.25 mA = 3.5 mV.

So the zener diode ("internal" Zener diode + \$R_Z\$) will drop a total of 14.762 V + 3.5 mV = 14.7655 V

Now we can determine the voltage across resistor \$R\$: 24 V - 14.7655 V = 9.2345 V and as the current will be 0.25 mA, \$R\$ will have a value of 9.2345 / 0.25 mA = 36.938 kohm

you can calculate Rmax but then the slope will change and you cant assume Rz is 14 ohm right??? and this Vo in second case is larger than first case is wrong right??

The \$R_Z\$ = 14 ohms remains valid for the whole defined range of the current through the zener diode so \$I_{ZK} &lt; I_Z &lt; I_{Zmax}\$ = \$0.25 mA &lt; I_Z &lt; 100 mA\$. So the slope (\$V_Z / I_Z\$) will not change, as it is defined by \$R_Z\$ which is constant.