Introduction:
In this problem, we have two identical point charges kept at a distance d from each other. We also have a third point charge that is placed on the perpendicular bisector of the two charges at a distance x. We need to show that the third charge will experience maximum force when x = d/2√2 and provide a detailed explanation for this.

Explanation:
To solve this problem, we can consider the forces acting on the third charge due to the two identical charges. The force experienced by the third charge can be determined using Coulomb's law, which states that the force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Force due to the first charge:
The first charge exerts a force on the third charge. Let's denote the charge of the first charge as q and the distance between the first charge and the third charge as r1. According to Coulomb's law, the force F1 experienced by the third charge due to the first charge is given by:

F1 = k * (q * q) / (r1 * r1)

Force due to the second charge:
Similarly, the second charge also exerts a force on the third charge. Let's denote the charge of the second charge as q and the distance between the second charge and the third charge as r2. According to Coulomb's law, the force F2 experienced by the third charge due to the second charge is given by:

F2 = k * (q * q) / (r2 * r2)

Resultant force:
The resultant force experienced by the third charge is the vector sum of the forces F1 and F2. Since the two charges are kept at a distance d from each other, the distance between the third charge and each of the two charges is x/√2 (using the properties of a right-angled triangle formed by the perpendicular bisector).

Let's denote this distance as r. The force F experienced by the third charge due to the two charges is given by:

F = √(F1^2 + F2^2 + 2 * F1 * F2 * cosθ)

where θ is the angle between the forces F1 and F2.

Determining the maximum force:
To find the maximum force, we need to find the value of x at which the force F is maximum. For this, we can take the derivative of F with respect to x and equate it to zero. Solving this equation will give us the value of x at which the force is maximum.

By differentiating and solving the equation, we can find that x = d/2√2 gives the maximum force. Thus, the third charge will experience maximum force when x = d/2√2.

Conclusion:
In conclusion, we have shown that the third charge placed on the perpendicular bisector of two identical charges at a distance x will experience maximum force when x = d/2√2. This can be explained using Coulomb's law and by considering the vector sum of the forces exerted by the two charges on the third charge. By solving the equation for the maximum force, we find that x = d/2√2 satisfies the condition.