class: center, middle, inverse, title-slide .title[ # Lagrange Multipliers ] .subtitle[ ## Session 08 ] .author[ ### Alejandro Ucan ] .date[ ### 2024-03-02 ] --- # Goals for the session: * Understand the necessary and sufficient conditions for the existence of extreme values in problems with restriction. <br/><br/> * Find these values using the Lagrange multiplier method. --- # Restrictions? > A rectangular box without a lid is made from `\(12m^2\)` of cardboard. Find the maximum volume of the box. <br/> -- Note that we have a restriction on the variables and it is that they satisfy `$$A_{sup}=2xz+2zy+xy=12.$$` --- # Intuitively We want to find the extreme values of a function `\(f(x,y)\)` such that the variables satisfy a restriction that can be expressed as a level curve of a function `\(g(x,y).\)` <br/><br/> We want to find the `\((x,y)\)` such that `$$f(x,y)\mbox{ has an extreme value}$$` `$$g(x,y)=k.$$` --- ## Visually <iframe src="https://www.geogebra.org/classic/uebb3j98?embed" width="1200" height="450" allowfullscreen style="border: 1px solid #e4e4e4;border-radius: 4px;" frameborder="0"></iframe> --- ## The multiplier > **Definition:** Given the conditions of the previous drawing, we will say that `\(\lambda\)` is a **Lagrange multiplier** if it satisfies that `$$\nabla f(x_0,y_0,z_0)=\lambda \nabla g(x_0,y_0,z_0).$$` --- # The Lagrange multiplier method > To find the extreme values of a function `\(f(x,y,z)\)` such that the variables satisfy a restriction that can be expressed as a level curve of a function `\(g(x,y,z).\)` We can use the following steps: * Find the values `\(x,y,z\)` and `\(\lambda\)` such that `$$\nabla f(x,y,z)=\lambda \nabla g(x,y,z).$$` * Evaluate `\(f\)` at these points, the largest of these numbers is the maximum and the smallest the minimum. --- # Ejemplo 1: > Find the extreme values of the function `\(f(x,y)=5x^2+6y^2-xy\)` under the restriction `\(x+2y=24.\)` -- 1. Compute the gradient of the objective function: `$$\nabla f= (10x-y,12y-x)$$` 1. Compute the gradient of the restriction: `$$\nabla g =(1,2)$$` 1. Find the values of `\(x,y\)` and `\(\lambda\)` such that `\(\nabla f(x,y)=\lambda \nabla g(x,y).\)` `$$10x-y=\lambda$$` `$$12y-x=\lambda$$` `$$x+2y=24$$` 1. Solve the system of equations. 1. Explain my results `$$y=-\frac{-288}{25},\quad x=\frac{576}{5}.$$` --- # Example 2: > The Cobb-Douglas production function for a production is given by `\(f(x,y)=100x^{3/4}y^{1/4}.\)` Where `\(x\)` are the units of production (at 150mxn per unit) and `\(y\)` are the units of capital (at 250mxn per unit). If our production and capital units are restricted to 50000mxn. Find the maximum production for this manufacturing. --- # Ejemplo 3: > BioTech is a manufacturer of prothesis and has developed a model of earnings that depends on the number of prothesis ($x$) that sells per month (in thousands) and the number of hours per month that invests in advertising ($y$). `$$f(x,y)=48x+96y-x^2-2xy-9y^2.$$` Our budget for production of prothesis and advertising is given by `\(20x+4y=216.\)` Find the number of prothesis and hours of advertising that maximize my earnings. --- # Ejemplo 4: > Find the extreme values for the function `\(f(x,y,z)=2x^2+y^2+3z^2\)` under the restriction `\(2x-3y-4z=49.\)`