You may use the applet to locate, by moving the little circle on the parabola, the extrema of the objective function along the constraint curve [math]g\left(x,y\right)=0[/math]. According to the method of Lagrange multipliers, an extreme value exists wherever the normal vector to the (grey) level curves of [math]f\left(x,y\right)[/math] and the normal vector to the (blue) constraint curve are parallel (or coincide on the graph).