I would like to conclude that $Ke^{f}=1$ on $\partial M$ from the following:
We have $M$ is a compact Riemannian manifold with boundary $\partial M$, $f\in C^{\infty}(\overline{M})$, $K$ is a constant. For $\varphi\in C^{1}(\overline{M})$, we have:
$$\int_{\partial M}\varphi^{2}(Ke^{f}-1)d\sigma=0.$$
Then, since $\varphi$ is arbitrary in $C^{1}(\overline{M})$, I would like to use the Fundamental Lemma of Calculus of Variations to conclude that $Ke^{f}-1=0$ on $\partial M$. However, I am not sure if I can use such Lemma. Can I?
I also have that $Ke^{f}-1\geq0$ on $\partial M$, which has allowed me to conclude that $\varphi^{2}(Ke^{f}-1)=0$ almost everywhere on $\partial M$, but from this I believe, I cannot conclude that $(Ke^{f}-1)=0$ as I need. That is my guess, but I can be wrong.
Any help, would be appreciated.