Suppose that we have a function, $f:\mathcal{X} \rightarrow \mathbb{R}$, with $\mathcal{X}=[a,b]\subset\mathbb{R}$, and a functional, $F:\mathbb{R}^{\mathcal{X}} \rightarrow \mathbb{R}$, and it is Fréchet differentiable. The functional has a local extremum at $f^*$, so the first derivative on $f^*$ is zero. From here, I found the Taylor series for the functional to be $$ F[f^*+\eta] = F[f^*] + \sum_{n=1}^{\infty} \frac{1}{n!}\int_a^b \frac{\delta^nF[f^*]}{\delta f^*(x_1) ... \delta f^*(x_n)}\ \eta(x_1) ... \eta(x_n)\ \mathrm{d}x_1...\mathrm{d}x_n. $$
Using Lagrange error bound, I found an equation below $$ \left| F[f^*+\eta] - F[f^*] \right| = \frac{1}{2} \left| \int_a^b \frac{\delta^2J[f^*+\varepsilon\eta]}{\delta f^*(x_1) \delta f^*(x_2)}\ \eta(x_1) \eta(x_2)\ \mathrm{d}x_1 \mathrm{d}x_2 \right|. $$ for some $0 \leq \varepsilon \leq 1$. The first derivative of $F$ on $f$ is zero because of the local extremum condition.
I would like to know if the bound from the equation above can take a form of $$ \left| F[f^*+\eta] - F[f^*] \right| \leq M \left| \int_a^b \eta(x) \mathrm{d}x \right|^2 $$
where $M$ does not depend on $\eta$. It is OK for $M$ to depend on $J[f^*]$ and its derivatives at some points.
Any help or hint will be appreciated!