Answer is: option4

Solution:

1. The horizontal distance between two points (x₁, y₁) and (x₂, y₂) is |x₁ − x₂|.
2. For the functions f(x) = x and g(x) = x², we want to find x such that the horizontal distance between f and g is maximized.

For a given y, solve for x in both functions:

1. From f(x) = y, we get x = y.
2. From g(x) = y, we get x = √y (since x ≥ 0).

The horizontal distance D(y) is:

D(y) = √y − y

Note: Since √y ≥ y for 0 ≤ y ≤ 1, we don't need the absolute value.

Find the maximum of D(y) = √y − y for 0 ≤ y ≤ 1.

Compute the derivative of D(y) with respect to y:

D'(y) = 1 / (2√y) − 1

Set the derivative equal to zero to find critical points:

1 / (2√y) − 1 = 0    ⟹    1 / (2√y) = 1    ⟹    √y = 1/2    ⟹    y = 1/4

Verify the maximum by checking the endpoints and the critical point:

1. At y = 0: D(0) = 0 − 0 = 0
2. At y = 1/4: D(1/4) = √(1/4) − 1/4 = 1/2 − 1/4 = 1/4
3. At y = 1: D(1) = 1 − 1 = 0

The maximum horizontal distance is 1/4.

The correct answer is D.