MAXIMA AND MINIMA

ARTICLE -1

Definite,Semi-Definite and Indefinite Function

DEFINITE FUNCTION A real valued function f with domain is said to be positive definite if f(x)>0 and negative definition if f(x)<0

EXAMPLE  The function defination by is positive definite. Here  The function defined by is negative definite. Here

NOTE A positive definite or a negative definite function is said to be a definite function.

Semi-Definite  A real valued function f with domain is said to be semi-definite if it vanishes at some points of and when it is not zero, it is of the same sign throughout

EXAMPLE  The function defined by is semi definite as and when then where

INDEFINITE FUNCTION  A real valued function f with domain is said to be indefinite if it can take values which have different signs i.e., it is nether definite nor semi definite function. is said to be indefinite if it can take values which have different signs i.e., it is nether definite nor semi definite function.

EXAMPLE  The function defined by is an indefinite function. Here can be positive, zero or negative.

ARTICLE -2

CONDITIONS FOR A DEFINITE FUNCTION

Quadratic Expression of Two Real Variables Let Let

cases

CASES- 1 If and,then f is positive semi-definite.

CASES- 2 If and,then f is negative semi-definite.

CASES- 3 If,then f(x,y) can be of any sign. f is indefinite.

QUADRATIC EXPRESSION OF THREE REAL VARIABLES  Let When then If are all positive f is positive semi-definite. f is positive semi-definite. When,then if the above three expression are alternately negative and positive. f is negative semi-definite. f is negative semi-definite.

ARTICLE -3

MAXIMUM VALUE A function f(x,y) is said to have a maximum value at x=a, y=b if f(a,b)>f(a+h,b+k) for small values of h and k, positive or negative.

MINIMUM VALUE A function f(x,y) is said to have a maximum value at x=a, y=b if f(a,b)<f(a+h,b+k) for small values of h and k, positive or negative.

EXTREME VALUE A maximum or a minimum value of a function is called an extreme value.

NOTE and are necessary but not sufficient conditions.

ARTICLE -4

WORKING METHOD FOR MAXIMUM AND MINIMUM Let f(x,y) be given functions

STEPS

STEP-1 Find and

STEP-2 Solve the equations and simultaneously for x and y. Solve the equations and simultaneously for x and y. Let be the points Let be the points

STEP-3 Find and calculate values of A,B,C for each points.

STEP-4 If for a point,we have and then f(x,y) is a maxima for this pair and maximum value is If for point,we have and then f(x,y) is a minimum for this pair and minimum value is If for point then there is neither max. nor minimum of f(x,y). In this case f(x,y) is said to have a saddle at

If for some point (a,b) the case is doubtful If for some point (a,b) the case is doubtful In this case, if for small values of h and k, positive or negative, then f is max. at (a,b). if for small values of h and k, positive or negative, then f is max. at (a,b). if for small values of h and k, positive or negative,then f is min. at (a,b). If dose not keep the same sign for small values of h and k, then there is neither max.nor minimum value.

NOTE The point are called stationary or critical points and values of f(x,y) at these points are called stationary values.

LAGRANGE’S METHOD OF UNDETERMINRD MULTIPLERS

Let f(x,y,z) be a function of x,y,z which is to be examied for maximum or minimum value and let the variable be connected by the relation …….(1) Since f(x,y,z) is to have a maximum or minimum value

Multiplying(2) by (3) by and adding, we get, For this equation to be satisfied identically, coeffs. of dx,dy,dz should be separately zero. Equation (1),(4),(5) and(6) give us the value of x,y,z, for which f(x,y.z)is maximum and minimum.