Functionals with constraints

Question

Say we wish to extremize the functional $$J[y]=\int_{\alpha}^{\beta}f[x,y(x),y'(x)]dx$$ under the constraint $$K[y]=\int_{\alpha}^{\beta}g[x,y(x),y'(x)]dx=C$$ where $C$ is a constant. How can one prove that the optimal solution extremizes the functional $J[y]-\lambda K[y]$ for any choice of $\lambda$?