Given Circuit:
The given circuit consists of a voltage source (V), a resistor (R), an inductor (L), and a switch (K). Switch K is initially open and is closed at t = 0.

Analysis:
To find the value of current (i), di/dt, and d2i/dt2 at t = 0, we need to analyze the circuit using Kirchhoff's voltage law (KVL) and the basic equation for an RL circuit.

Kirchhoff's Voltage Law (KVL):
According to KVL, the sum of voltages across all elements in a closed loop is equal to zero. In the given circuit, the loop consists of the voltage source, resistor, and inductor. Therefore, we can write:

V - iR - L(di/dt) = 0

This equation represents the voltage drop across the elements in the loop.

Basic Equation for an RL Circuit:
The basic equation for an RL circuit relates the voltage across an inductor to the rate of change of current flowing through it. It can be written as:

V = L(di/dt) + Ri

This equation represents the voltage drop across the inductor.

Equating the Equations:
By equating the above two equations, we can eliminate the voltage source and derive a differential equation for the current (i) in the circuit:

iR + L(di/dt) = L(di/dt) + Ri

Simplifying the equation, we get:

iR = L(di/dt)

Solving the Differential Equation:
To solve the differential equation, we can rearrange it as:

di/i = -R/L dt

Integrating both sides, we get:

ln|i| = -R/L t + C

Where C is the integration constant.

Applying Initial Condition:
At t = 0, the switch K is closed, and the current starts to flow. Therefore, the initial condition is i(0) = 0.

Substituting the initial condition into the equation, we get:

ln|0| = -R/L * 0 + C
C = ln|0|

However, ln|0| is undefined, so C remains undetermined.

Conclusion:
In conclusion, at t = 0, the value of the current (i) in the circuit is 0. The derivative of current, di/dt, at t = 0 cannot be determined without knowing the constant C. Similarly, the second derivative of current, d2i/dt2, at t = 0 cannot be determined without the value of C. Therefore, without additional information or constraints, we cannot find the exact values of di/dt and d2i/dt2 at t = 0.