Consider the following functional: $$ F[y] = \int_{x_1}^{x_2}f(y,\dot{y}, x) dx $$ The variation can be expanded to second order as following, where we also assume that variations vanish $\delta y$ vanish at the boundaries and that the path $x$ is an extremum and annihilates the first order terms. We use integration by parts to go from the second line to the third line: \begin{align*} \delta F &= \int_{x_1}^{x_2}f(y+\delta y,\dot{y} + \delta{\dot{y}}, x) dx - \int_{x_1}^{x_2}f(y,\dot{y}, x) dx \newline &= \frac{1}{2}\int_{x_1}^{x_2}\bigg[ \frac{\partial^2 f}{\partial y^2} \delta y \delta y + 2 \frac{\delta^2 f}{\partial y \partial \dot{y}}\delta y\delta \dot{y} + \frac{1}{2}\frac{\partial^2 f}{\partial \dot{y}^2} \delta{\dot{y}}\delta{\dot{y}}\bigg]\newline &= \frac{1}{2}\int_{x_1}^{x_2}\bigg[ \frac{\partial^2 f}{\partial y^2} \delta y \delta y - \frac{d}{dt}\bigg(\frac{\partial^2 f}{\partial y \partial \dot{y}}\bigg) \delta y \delta y + \frac{\partial^2 f}{\partial \dot{y}^2} \delta{\dot{y}}\delta{\dot{y}}\bigg] \end{align*}
Are there any theorems (in particular, iff statements) that explain any conditions on the derivatives that make the integral positive, negative, or zero?