A maximization problem in Sobolev space

Question

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ functions whose weak derivatives are also in $L^2(0,1)$ (One can simply view it as $C^1([0,1])$ with derivatives vanishing at the two ends, for convenience). This sequence $f_k$ satisfies the following properties: (1) it is strictly decreasing in $x$, that is, for any $k$, $f_k^{'}(x)<0, \forall x\in (0,1)$; (2) $f_k^{'}(0)=0=f_k^{'}(1)$; (3) $\int_0^1f_k(x)dx=2,\forall k$; (4) $f_k(0)=O(k)$ for sufficiently large $k\gg 1$ and (5) $f/2$ converges to a Delta function concentrated at zero as $k\rightarrow \infty$; more specifically $\int_0^1f_k(x)\phi(x)dx\rightarrow 2\phi(0)$ as $k\rightarrow \infty$ for any nice test function $\phi$. We now consider the general maximization problem: \begin{equation} \lambda_k=\max_{w\in H^1_N(0,1),w\neq 0}\frac{-\int_0^1(|w^{'}|^2+w^2)dx+k \Bigr(\int_0^1f_kw^2dx-\frac{(\int_0^1f_kwdx)^2}{\int_0^1f_kdx}\Bigr)}{\int_0^1w^2dx}. \quad (MPG) \end{equation} By Holder inequality, the second term in the numerator is non-negative. Also, by choosing $w=1$, we immediately see $\lambda_k\geq -1$ for all $k$. The general question is however to ask whether $\lambda_k>0$ for all large $k\gg1$ or not, or can we determine the sign of $\lambda_k$ for all large $k$. I cannot figure out this general question. The following are some trials. I spent lots of efforts finding such a sequence: let $$g_k(x)=ke^{-\frac{k^2}{2}x^2}+\frac{k^3}{2}e^{-\frac{k^2}{2}}x^2. \tag{G} $$ Then by using properties of the normal density $N(0,1/k^2)$ one has $$ s_k:=\int_0^1g_k(x)dx=k\int_0^1e^{-\frac{k^2}{2}x^2}dx+\frac{k^3}{6}e^{-\frac{k^2}{2}}\rightarrow \sqrt{\pi/2}. \tag{S} $$ Then one can check $h_k(x)=2g_k(x)/s_k$ is positive and satisfies (1)-(5). Substituting this into (MPG), upon some eliminations, we get a specific maximization problem: \begin{equation} \mu_k=\max_{w\in H^1_N(0,1),w\neq 0}\frac{-s_k^2\int_0^1(|w^{'}|^2+w^2)dx+2k \Bigr[s_k\int_0^1g_kw^2dx-(\int_0^1g_kwdx)^2\Bigr]}{s_k^2\int_0^1w^2dx}, \quad (MPS) \end{equation} where $g_k$ is defined by (G) and $s_k$ is defined by (S). Again, we are wondering whether $\mu_k>0$ for all $k\gg 1$ or not, or can we determine the sign of $\mu_k$ for all large $k$? Still it is not easy to see the answer, since the involved integrals are hard to evaluate. The starting candidate test functions are these Neumann eigenfunctions $e_n(x)=\cos(n\pi x), n\geq 0$, since they form a base of $H_N^1(0,1)$. For these functions, we have $$ \int_0^1(|e_0^{'}|^2+e_0^2)dx=1,\quad\int_0^1(|e_n^{'}|^2+e_n^2)dx=\frac{1+n^2\pi^2}{2}, n\geq 1, \quad \int_0^1x^2\cos(n\pi x)dx=\frac{2}{n^2\pi^2}(-1)^n, $$ $$ \int_0^1g_ke_ndx=k\int_0^1e^{-\frac{k^2}{2}x^2}\cos(n\pi x)dx+\frac{(-1)^nk^3}{n^2\pi^2}e^{-\frac{k^2}{2}} $$ and $$ \int_0^1g_ke_n^2dx=\frac{s_k}{2}+\frac{k}{2}\int_0^1e^{-\frac{k^2}{2}x^2}\cos(2n\pi x)dx+\frac{k^3}{8n^2\pi^2}e^{-\frac{k^2}{2}}. $$ Now, the issue is to calculate the Gauss-type integrals. I am not quite familiar in using software to do symbolic calculations. By technology or something else can somebody give me an answer? Thank you so much!