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The answer becomes clear if we choose $b$ and $\Delta b$ to saturate the bound.  In order to do this we must change our inequalities to equalities in part (a).  That is, we must have $\Vert b \Vert_2 = \Vert A \Vert_2 \Vert x \Vert_2 = \sigma_1 \Vert x \Vert_2$, which requires that $x$ **must be a multiple of the first right-singular vector** $v_1$ and hence $b$ is a **multiple of the first left-singular vector** $u_1$.