= Solution
For $0<\alpha<\beta$, abbreviate $D_\alpha=D(u_\alpha)$ and $J_\alpha=J(u_\alpha)$. Optimality of the two <variational regularization> solutions gives
$$
D_\alpha+\alpha J_\alpha\leq D_\beta+\alpha J_\beta,\qquad
D_\beta+\beta J_\beta\leq D_\alpha+\beta J_\alpha.
$$
Adding shows $(\beta-\alpha)(J_\beta-J_\alpha)\leq0$, so $J_\beta\leq J_\alpha$. Rearranging each comparison then gives the <monotonicity of data fidelity and regularization penalty>
$$
\boxed{0\leq\alpha(J_\alpha-J_\beta)
\leq D_\beta-D_\alpha
\leq\beta(J_\alpha-J_\beta).}
$$
Hence \b[the regulariser value is non-increasing and the data misfit is non-decreasing with the parameter]. These inequalities hold for any choices of minimizers at the two distinct parameters, including when uniqueness fails.
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