To find the work done when a wire is stretched by an amount x, we need to calculate the potential energy stored in the wire. The potential energy stored in a material when it is stretched or compressed is given by the formula:

Potential Energy = (1/2) * (Stress) * (Strain) * (Volume)

In this case, the stress is given by the equation:

Stress = (Force) / (Area)

And the strain is given by the equation:

Strain = (Change in length) / (Initial length)

Let's break down the solution into steps:

Step 1: Calculate the stress:
Stress = (Force) / (Area)
Since the wire is stretched, the force applied is equal to the force required to stretch the wire. This force can be calculated using Hooke's Law:

Force = (Young's modulus) * (Area) * (Strain)

Substituting this into the stress equation, we get:
Stress = ((Young's modulus) * (Area) * (Strain)) / (Area)
Stress = Young's modulus * Strain

Step 2: Calculate the strain:
Strain = (Change in length) / (Initial length)
In this case, the change in length is given as x and the initial length is given as L.

Strain = x / L

Step 3: Calculate the potential energy:
Potential Energy = (1/2) * (Stress) * (Strain) * (Volume)
The volume of the wire is given by the equation:
Volume = (Area) * (Length)
Substituting the stress and strain equations, we get:
Potential Energy = (1/2) * (Young's modulus * Strain) * (Strain) * (Area * Length)
Potential Energy = (1/2) * (Young's modulus * Strain^2) * (Area * Length)
Potential Energy = (1/2) * (Young's modulus) * (x^2 / L^2) * (Area * Length)

Step 4: Simplify the equation:
Potential Energy = (1/2) * (Young's modulus * x^2 * Area * Length) / (L^2)
Potential Energy = (Young's modulus * x^2 * Area * Length) / (2 * L^2)
Potential Energy = (2 * Young's modulus * x^2 * Area * Length) / (2 * L^2)
Potential Energy = (2 * Young's modulus * x^2 * Area) / (2 * L)
Potential Energy = (Young's modulus * x^2 * Area) / L

Step 5: Simplify the equation further:
Potential Energy = Young's modulus * (x^2 * Area) / L
Potential Energy = x^2 * (Young's modulus * Area) / L
Potential Energy = x^2 * Y * A / L

So, the work done when a wire is stretched by an amount x is given by the equation:
Work Done = x^2 * Y * A / L

Therefore, the correct answer is option C: x^2 * Y * A / 2L.