Why there is no current in the inductor during the first positive half cycle of the ac voltage?
The waveform is wrong. You are correct, the current should rise during the first half cycle. When the power is switched on, there is an initial response that is different from the steady state response. That is not shown in this case.

To see what it looks like for switch-on at voltage zero crossing, look at the answer by @Andy aka here: What is the difference beteeen a solenoid&#39;s make current and inrush current?

Image from that answer:



The current starts to rise from zero as soon as the voltage begins to rise from zero. Since there is no reverse current at the start, the current rises above the normal steady state AC current. It keeps rising until the voltage reaches zero. Since the current rose higher than normal, the reverse voltage does not take the negative peak as low as it will at steady state. The current is a sine wave, but it has a DC offset that lasts for some period of time. The decay of the offset assumes that this is a real circuit with losses rather than and ideal circuit with no losses as shown in the diagram question.