Question #40303

This problem tests the 'Universal Set Relativity' and 'Constant Function Logic' by evaluating the boundary conditions of strict monotonicity.

Students often assume that a system of linear constraints on a polynomial's coefficients will yield a valid function family, focusing only on the algebraic substitution.

The trap lies in the strict monotonicity condition $f'(x) \ge 0$, which translates to a quadratic inequality in $x^2$.

By setting the boundary conditions using specific cognitive load anchors (262 and 20), the problem forces the vertex of the derivative parabola into the positive domain ($t_v > 0$).

Students who fail to check the global minimum on the restricted domain $t \ge 0$, or who incorrectly assume $f'(x) > 0$ implies a simple global discriminant condition ($\Delta \le 0$ for the biquadratic), will falsely conclude a function exists.

The negative discriminant of the final constraint proves the function set effectively 'leaves' the universal set of strictly increasing functions, resulting in an empty set.