# Errors and Approximations, Maxima and Minima, Lagrange’s Method

• ### Errors and Approximations

If ${f}$ is a function of ${x,y,z}$, then the error in ${f}$ is

${df = \frac {\partial f}{\partial x} dx + \frac {\partial f}{\partial y} dy + \frac {\partial f}{\partial z} dz}$

However, this is an approximation.

Note: If necessary, we take log on both sides.

• ### Maxima and Minima (Functions of Several Variables)

We have studied the process of finding extreme values of ${y=f(x)}$. Now, we will see the same for ${f = f(x,y)}$.

First, we equate ${\frac {\partial f}{\partial x}}$ and ${\frac {\partial f}{\partial x}}$ to ${0}$. We get those pairs ${(x,y)}$, which can either be maxima, minima or saddle points.

We then find ${f_{xx} = r}$, ${f_{xy} = s}$ and ${f_{yy}=t}$ for these pairs.

I) ${rt > s^2}$, ${r < 0}$ – Gives maximum value

II) ${rt > s^2}$, ${r > 0}$ – Gives minimum value

III) ${rt < s^2}$Saddle point

IV) ${rt = s^2}$ – No conclusion possible

• ### Lagrange’s Method of Undetermined Multipliers

This method is used when the maximum and minimum of a function are to be obtained under certain constraints (like the optimization problems). We form a linear relation between the function and the constraint using a Lagrange multiplier and differentiate it partially w.r.t. the variables. We then eliminate the variables and parameter and get an equation. The roots of that equation are the extreme values.