Utility Representation and Positive Monotonic Transform: A Proof for Maria's Preferences

To show that f(u) = 1/3u^2 is a positive monotonic transform of u = √(3x + 3y), we need to demonstrate that it satisfies two conditions:

Monotonicity: This means that if u1 > u2, then f(u1) > f(u2). To prove this, let's assume u1 > u2.

u1 = √(3x1 + 3y1) > √(3x2 + 3y2) = u2

Squaring both sides, we get:

3x1 + 3y1 > 3x2 + 3y2

Dividing both sides by 3:

x1 + y1 > x2 + y2

Multiplying both sides by 1/3 and squaring:

(1/3)(x1 + y1)^2 > (1/3)(x2 + y2)^2

Substituting u1 and u2 back into f(u):

f(u1) = 1/3u1^2 = 1/3(3x1 + 3y1) = x1 + y1

f(u2) = 1/3u2^2 = 1/3(3x2 + 3y2) = x2 + y2

Therefore, we have shown that f(u1) > f(u2) when u1 > u2, confirming that f(u) is a monotonic function.
Positivity: This means that f(u) > 0 for all u > 0. To prove this, we substitute u = √(3x + 3y) into f(u) and simplify:

f(u) = 1/3u^2 = 1/3(3x + 3y) = x + y

Since Maria values tennis matches and karaoke sessions equally, we know that x = y. Therefore, f(u) = 2x > 0 for all x > 0, proving that f(u) is a positive function.

Since f(u) satisfies both conditions, we have successfully demonstrated that it is a positive monotonic transform of u = √(3x + 3y).