RLC circuit: how to find voltage/current derivatives and derive differential equation
Capacitor and Inductor are devices capable of storing energy in electrostatic and electromagnetic fields, respectively due to which, in simple words, the voltage across the capacitor and the current through inductor cannot change instantly. This is what you recognised as axioms.

Let's see, 
What happens at t=0?

The current source is 'open' (50u(-t)=0 at t=0 and for all t>0).
The voltage source is now introduced (which was earlier 'shorted' as u(t) =0 for t&lt;0).
Due to these changes the previous steady state conditions are disturbed and new steady state conditions will be reached after transient. 
During transient at t=0+, iL(0-)=iL(0+) and vC(0-)=vC(0+). 

At t=0+, as current source is out (Note that it is not neglected here, it IS 0 ! ), remaining elements will be in series and following the required condition that iL(0+)=iL(0-), the current through all elements at t=0+ should be 0. It can't be anything else because if it were, it'd have violated iL(0-)=iL(0+)=0.

(In response to his comments) 
So why you can't write by Ohm's law as current=voltage /sum of resistance? Because in Ohm's law, the voltage should be voltage across resistance(s), which here is not the source voltage (can you find out what is it and then apply Ohm's law to check whether you're getting 0 or not? It'd strengthen your skills! HINT: Voltage across inductor would be Ldi/dt, terminal at which current enters being more positive).