Answer to Question #104764 in Calculus for maria

Question

How to solve a multivariable function using Lagrange multipliers

Answer 1

I will talk about the function of two variables, but all that has been said will be correct for more variables.

Suppose we want to find the conditional extremum of the function $f(x,y)$ with the condition $g(x,y)=0$ . To do this, construct a new function

$$F(x,y;\lambda)=f(x,y)-\lambda\cdot g(x,y)$$

Next we use this new function $F(x,y;\lambda)$ and try to find the stationary points of the point from the necessary condition for the extremum of the function

$$\left\{\begin{array}{l} F'_x(x,y;\lambda)=0\\ F'_y(x,y;\lambda)=0\\ F'_\lambda(x,y;\lambda)=0\\ \end{array}\right. \\\longrightarrow (x_n,y_n;\lambda_n)$$

are solutions of the system of equations.

Using the Lagrange multiplier method, we can obtain sufficient conditions for a conditional extremum that require analysis (in the simplest case) of the second derivatives of the Lagrange function $F(x,y;\lambda).$