MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Let $\lambda_1,\ldots,\lambda_m$ real numbers pairwise distinct and $\mu_1,\ldots,\mu_m$ real numbers all nonzero. We know from the Lagrange polynomial interpolation that there exists an unique polynomial $R$ of degree less than $(m+1)$ such that $R(\lambda_i)=\mu_i,1\leq i\leq m$. We can prove by the means of an adequate use of the intermediate value Theorem, the existence of a real $\alpha$ such that the polynomial $P_{\alpha}(x)=R(x)+(x-\alpha)(x-\lambda_1)\ldots(x-\lambda_m)$ admits $(m+1)$ real roots pairwise distinct.
I wonder if it is possible to generalize this property. Precisely, let $1\leq r\leq m$, we denote by $Q$ the polynomial of degree less than $(m+r+1)$ such that $Q(\lambda_i)=\mu_i,1\leq i\leq m$ and $Q'(\lambda_i)=0,1\leq i\leq r$. Is it possible to find $(\alpha,\beta)\in\mathbb{R}^2$ such that the polynomial $$S_{\alpha,\beta}(x)=Q(x)+(x-\alpha)(x-\beta)\displaystyle\prod_{i=1}^{r}(x-\lambda_i)^2\displaystyle\prod_{i=r+1}^{m}(x-\lambda_i)$$ admits $(m+r+2)$ real roots pairwise distinct?