Introduction:
In this problem, we are given a cantilever beam of constant depth carrying a uniformly distributed load on the entire span. We need to determine the relationship between the breadth of the section at a distance 'x' from the free end and the maximum stress at all sections.

Explanation:
To understand why the breadth of the section should be proportional to x^2 in order to make the maximum stress the same at all sections, we need to consider the bending moment and the stress distribution along the cantilever beam.

Bending Moment:
The bending moment in a cantilever beam with a uniformly distributed load is given by the equation:

M = w * x^2 / 2

Where:
M = Bending moment at a distance 'x' from the fixed end
w = Uniformly distributed load per unit length
x = Distance from the fixed end

Stress Distribution:
The maximum stress in a beam occurs at the section farthest from the neutral axis. In the case of a cantilever beam, the maximum stress occurs at the fixed end. The stress distribution across the beam is parabolic, and the maximum stress is given by:

σ_max = M * y / I

Where:
σ_max = Maximum stress
M = Bending moment at the section
y = Distance of the section from the neutral axis
I = Moment of inertia of the section

Key Points:
- The bending moment increases as we move away from the free end of the cantilever beam.
- The maximum stress occurs at the section farthest from the neutral axis, which is the fixed end.
- To have the same maximum stress at all sections, the moment of inertia of the sections should be the same.

Derivation:
Let's consider two sections at distances 'x1' and 'x2' from the free end, where x2 > x1.

For both sections, the maximum stress should be the same. Therefore, we have:

M1 * y1 / I1 = M2 * y2 / I2

Since the maximum stress occurs at the section farthest from the neutral axis (fixed end), we have:

y1 = h/2
y2 = h

Where:
h = Depth of the section

Substituting these values into the equation, we get:

M1 * h / (2 * I1) = M2 * h / I2

Since the depth is constant, we can cancel it out:

M1 / (2 * I1) = M2 / I2

Using the equation for bending moment, we have:

w * x1^2 / (2 * I1) = w * x2^2 / I2

Rearranging the equation, we get:

x2^2 / I2 = x1^2 / (2 * I1)

Since we want the maximum stress to be the same at all sections, we need the moment of inertia to be the same. Therefore, we have:

I1 = I2

Substituting this into the equation, we get:

x2^2 = x1^2 / 2

Taking the square root of both sides, we get:

x2 = √(x1^2 / 2)

Therefore, the breadth of the section at a distance 'x'