Maximum Deflection of a Cantilever
The maximum deflection of a cantilever beam occurs at the free end of the beam when subjected to a load. The deflection can be calculated using the principles of mechanics and beam theory.

Flexural Rigidity (EI)
Flexural rigidity (EI) is a property of a beam that represents its resistance to bending. It is the product of the Young's modulus (E) and the moment of inertia (I) of the beam's cross-sectional area. The flexural rigidity determines how much a beam will deflect under a given load.

Calculation of Maximum Deflection
To calculate the maximum deflection of a cantilever beam, we can use the formula:

δ = (WL^3) / (3EI)

Where:
δ = Maximum deflection
W = Applied load at the free end of the cantilever
L = Length of the cantilever
E = Young's modulus of the material
I = Moment of inertia of the cross-sectional area

Explanation of the Formula
The formula for maximum deflection of a cantilever beam is derived from the differential equation of beam bending. It takes into account the applied load, length of the beam, flexural rigidity, and the mechanical properties of the material.

The factor of (WL^3) in the numerator represents the applied load multiplied by the cube of the length of the cantilever. This factor indicates that the deflection is directly proportional to the cube of the length and the applied load.

The factor of 3EI in the denominator represents the flexural rigidity of the beam. It indicates that the deflection is inversely proportional to the flexural rigidity. A higher flexural rigidity will result in a smaller deflection.

Conclusion
In conclusion, the correct formula for the maximum deflection of a cantilever beam under a load W at the free end is δ = (WL^3) / (3EI). This formula takes into account the length of the cantilever, applied load, and the flexural rigidity of the beam. By calculating the maximum deflection, engineers can ensure that the beam will not exceed its allowable limits and maintain structural integrity.