The Lagrange multiplier technique is how we take advantage of the observation made in the last video, that the solution to a constrained optimization problem.

LagrangeMultipliers2D - Lagrange multiplier - Wikipedia

Here, you can see a proof of the fact shown in the last video, that the Lagrange multiplier gives information about how altering a constraint can alter the s.

Ali Issa Estimating the demand for money in Libya: An application of the Lagrange multiplier structural break unit root test and the ARDL cointegration approach