Second order necessary and second order sufficient conditions are obtained for the unconstrained minimization of a functional \(f(x)=\int^{b}_{a}| g(x,t)| dt\). They are expressed in terms of the directional derivative, f’(x;h), and the curved second directional derivative, \[ f''(x:h,z)=\lim_{\lambda \to 0+}\lambda^{- 2}[f(x+\lambda h+\lambda^ 2z)-f(x)-\lambda f'(x;h)]. \] These derivatives are obtained explicitly for the given f(\(\cdot)\). An example is calculated.