Step-by-step Solution:

The temperature function is given as: \[ T(x, y) = 4x^2 - 4xy + y^2 = (2x - y)^2 \] The goal is to maximize the function \( 2x - y \) while satisfying the constraint \( x^2 + y^2 = 25 \), which is the equation of a circle with radius 5 centered at the origin. Let \( 2x - y = k \). The maximum value of \( k \) will occur when the line \( 2x - y = k \) is tangent to the circle \( x^2 + y^2 = 25 \). The condition for tangency is derived using the distance between the center of the circle \((0, 0)\) and the line \( 2x - y = k \): \[ c^2 = a^2 (1 + m^2), \] where \( c \) is the distance from the center to the tangent line, \( a \) is the radius of the circle, and \( m \) is the slope of the line. Here, the slope \( m \) of the line \( 2x - y = k \) is \( m = 2 \). Substituting into the tangency condition: \[ k^2 = 25(1 + 2^2), \] \[ k^2 = 25(1 + 4) = 125. \] Thus, the maximum value of \( k \) is: \[ k = \pm \sqrt{125} = \pm 5\sqrt{5}. \] Therefore, the maximum (or minimum) value of \( 2x - y \) is \( \pm 5\sqrt{5} \).