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The sets of admissible functions are nested:All impose the same left conditions; the right height adds a restriction, and the right slope adds another. Minimizing the same functional over a smaller set cannot lower its minimum. ThereforeThe doubly clamped beam has the greatest minimum energy, and the free-ended beam the smallest. No energy integrals are needed. In fact the inequalities are strict: the positive quadratic bending term makes the minimizer unique on the left-clamped space, and the three solutions in part (b) are different. The relevant principle is monotonicity of a constrained minimum under added constraints.