A family of positive convex function $\Psi_{\alpha} : \mathbb{R}^N \to \mathbb{R}$ is defined as $$\Psi_{\alpha}(x) = A(x) + B(x) + \alpha D(x)$$ with paramter $\alpha \in (0,\infty)$. $A(x),B(x),C(x)$ are all positive,quadratic and convex functions. (edit), Also $A,B,D$ are linearly independent.
Let the unique minimizer of $\Psi_{\alpha}(x)$ over $x \in \mathbb{R}^N$ be $x_{\alpha}$. Define $E(\alpha) = B(x_{\alpha})$. It is known that $\lim_{\alpha\to 0}E(\alpha) = 0$ and $\lim_{\alpha\to \infty}E(\alpha) = 0$.
I want to know if I can show that there exists a unique maximum of $E(\alpha)$, over $\alpha \in (0,\infty)$. Existence is clear. Uniqueness need to be proved.