I am constructing a smooth function $f(x)\equiv f(u(x),v(x))$, such that $u(x)$ and $v(x)$ are some trial parameters. I have the following integral $$G=\int_{x_i}^{x_f} f(u(x),v(x)) \mathrm{d}x.$$ My aim: To find the trial functions $u(x)$ and $v(x)$ using the stationary principle. Therefore, essentially I want $u(x)$ and $v(x)$ to be such that $G$ is minimum.
My question: Am I allowed to use: $$\frac{\partial f}{\partial u}=\frac{\partial f}{\partial v}=0$$ to solve for $u(x)$ and $v(x)$?