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.


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