For ill-posed equation
$$Ax=b $$
where $A\in\mathbb{R}^{m\times n}$ and $b=b_{ex}+e$ is perturbed by noisy vector $e$ with unknown exact data $b_{ex}=Ax_{ex}$ (suppose $\|e\|0$, we have
\begin{align}
\varphi(\lambda) = \sum_{i=1}^{n}\dfrac{\sigma_{i}^{2}(u_{i}^{T}b)^{2}}{(\sigma_{i}^{2}+\lambda)^{2}}+\|x_{ex}\|^{2}-2\sum_{i=1}^{n}\dfrac{\sigma_{i}(u_{i}^{T}b)(v_{i}^{T}x_{ex})}{\sigma_{i}^{2}+\lambda}.
\end{align}
and
\begin{align}
\varphi^{'}(\lambda) = 2\sum_{i=1}^{n}\dfrac{u_{i}^{T}b}{(\sigma_{i}^{2}+\lambda)^{3}}[(\sigma_{i}^{2}+\lambda)\sigma_{i}v_{i}^{T}x_{ex}-\sigma_{i}^{2}u_{i}^{T}b]
\end{align}
\textcolor{blue}{I want to find a sufficiently large value $C(A,b,e)$ such that if $\lambda>C(A,b,e)$, then $\varphi^{'}(\lambda)>0$.} If such $C(A,b,e)$ exists, this will means $\lambda^{*}\leq C(A,b,e)$. But I have difficulty deriving this. Could you help me to give an expression of $C(A,b,e)$ such that $\varphi^{'}(\lambda)>0$ for $\lambda>C(A,b,e)$?
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