Question #41455

This question tests the synchronization of domain constraints with integral partitioning and optimization.

The primary trap lies in the 'domain of validity' for logarithmic differentiation and exponential bases.

If a student ignores the absolute value and attempts to integrate or differentiate $(x-1)^x$ over $[0, 2]$, they violate the domain $x > 1$ required for real-valued variable-base exponentials, leading to undefined or complex regions.

By forcing the partition at $x=1$, the student must map both segments to a common domain $(0,1)$.

The subsequent step requires recognizing that $t^{1-t} + t^{1+t}$ can be factored to apply the AM-GM inequality, elegantly bypassing the need for intractable direct integration while strictly bounding the integral.