many variables

# Lecture ¹5. The maximum and minimum of function depending from many variables

Exercises for Lecture 5

The purpose of work:

To find the maximum and minimum of function depending from many variables.

Theoretical information and examples.

Content

1.13 The Extreme Value of a Function of Several Variables

1.13.1 Definition of an extreme value of a function

1.13.2 Necessary conditions for an extreme value

1.13.3 Sufficient conditions for an extreme value

1.13.4 The case of a function of many variables

1.13.5 Conditional extremum

1.13.6 Largest and smallest values of a function

1.14 Finding the Greatest and Smallest Values of Functions

1.13 The Extreme Value of a Function of Several Variables

1.13.1 Definition of an extreme value of a function. We say that a function f(x, y) has a maximum (minimum) f(a, b) at the point P(a, b), if for all points P'(x, y) other than P in a sufficiently small neighbourhood of P one has the inequality f(a,b) >f(x, y) [i.e., f(a, b) < f(x, y)]. The generic term for maximum and minimum of a function is extreme. In a similar fashion, we define the extreme value of a function of three or more variables.

1.13.2 Necessary conditions for an extreme value. The points at which a differentiable function f(x, y) may attain an extremum (so-called stationary points) are found by solving the system of equations:

(1)

(necessary conditions for an extremum). System (1) is equivalent to a single equation, df(x,y) = 0. In the general case, at the point of the extreme value P(a, b) of the function f(x, y), df(a, b) = 0 or df(a,b) does not exist.

1.13.3 Sufficient conditions for an extreme value. Let P(a, b) be a stationary point of the function f(x, y), i.e., df(a,b)=0. Then:

a) if d²f(a, b) < 0 for dx² + dy² > 0, then f(a, b) is a maximum of the function f(x, y),
b) if d²f(a, b) > 0 for dx² + dy² >0, then f(a, b) is a minimum of the function f(x, y),
c) if d²f(a, b) changes sign, then f(a, b) is not an extreme value of f(x, y).

These conditions are equivalent to the following: Let f 'x(a, b)=f 'y(a,b)=0 and A=f''xx(a,b),B=f''xy(a,b), C = f''yy(a,b). We form the discriminant

.

Then,

1) if, the function has an extremum at the point P(a, b), namely a maximum,

if A < 0 (or C < 0), and a minimum, if A> 0 (or C > 0)

2) if, then there is no extremum at P(a, b),

3) if, the question of an extremum of the function at P(a, b) remains open (i.e., it requires a further investigation).

1.13.4 The case of a function of many variables. For a function of three or more variables, the necessary conditions for the existence of an extremum are similar to Conditions (1), while the sufficient conditions are analogous to Conditions a), b), and c) of 1.13.3.

Example 1. Test for an extremum the function

Solution. Find the partial derivatives and form a system of Equations (1):

or

Solving the system, we get the four stationary points:

We find now the second derivatives

and form the discriminant for each stationary point.

1) For the point P1:

Thus, there is no extremum at this point.

2) For the point P2:

The function has a minimum which is equal to the value of the function for :

3) For the point P3:

There is no extreme value.

4) For the point P4:

At the point P4, the function has the maximum

1.13.5 Conditional extremum. In the simplest case, the conditional extremum of a function f(x, y) is a maximum or a minimum of this function which is attained on the condition that its arguments are related by the equation j(x,y) = 0 (coupling equation). In order to find the conditional extremum of a function f(x, y), given the relationship j(x, y) = 0, we form the so-called Lagrange function

where l is an undetermined multiplier, and we seek the ordinary extremum of this auxiliary function. The necessary conditions for the extremum reduce to the system of three equations

with the three unknowns x, y, l, from which it is, generally speaking, possible to determine these unknowns.

The question of the existence and character of a conditional extremurn is solved on the basis of a study of the sign of the second differential of the Lagrange function

for the given system of values of x, y, l obtained from (2) under the condition that dx and dy are interrelated by the equation

In fact, the function f(x, y) has a conditional maximum, if d²F < 0, and a conditional minimum, if d²F> 0. As a particular case, if the discriminant D of the function F(x, y) at a stationary point is positive, then there is at this point a conditional maximum of the function f(x, y), if A < 0 (or C < 0), a conditional minimum, if A > 0 (or C > 0).

In a similar fashion, we find the conditional extremurn of a function of three or more variables, provided there are one or several coupling equations (the number of which, however, must be less than the number of variables). Here, we have to introduce into Lagrange's function as many undetermined multipliers as there are coupling equations.

Example 2. Find the extremum of the function

provided the variables x and y satisfy the equation

Solution. Geometrically speaking, the problem reduces to finding the largest and smallest values of the z-co-ordinate of the plane

for points of its intersection with the cylinder

We form the Lagrange function

We have

The necessary conditions yield the system of equations:

Solving this system, we find

Since

it follows that

If l = 5/2, x = 4/5 and y = 3/5, then d²F > 0 and, consequently, the function has a conditional minimum at this point. If l=-5/2, x = -4/5 and y = -3/5, then d²F < 0, and, consequently, the function has at this point a conditional maximum.

Thus,

1.13.6 Largest and smallest values of a function. A function, which is differentiable in a limited closed region, attains its largest (smallest) value either at a stationary point or at a point of the boundary of the region.

Example 3. Determine the largest and smallest values of the function

in the region

Solution. This region is a triangle (Fig. 7).

1) We find first the stationary points:

whence x= -1, y= -1 and get the point M(-1, -1).

At M, the value of the function zM = -1. It is not absolutely necessary to test for an extremum.

2) Investigate the function on the boundaries of the region. When x = 0, z = y² + y and the problem reduces to seeking the largest and smallest values of this function of one argument on the interval We find that z(larger)x=0 = 6 at the point (0, -3), z(smaller)y=0 = -1/4 at the point (-1/2, 0).

When y = 0, we get z = x2+x. Similarly, we find that z(larger)y=0 = 6 at the point , z(smaller)y=0=-1/4 at the point (0,-1/2)

When x + y = - 3 or y = -3 - x, we will have z = 3x² + 9x + 6. Similarly, we find that z(smaller)x+y=-3=-3/4 at the point (-3/2, -3/2), z(larger)x+y=-3 coincides with z(larger)x=0 and z(larger)y=0. On the straight line x+y = -3, we could test the function for a condilional extreme value without reduction to a function of a single argument.

3) Correlating all the values of the function z obtained , we conclude that z(larger) = 6 at the points (0,-3) and (-3,0), z(smaller) = -1 at the stationary point M.

1.14 Finding the Greatest and Smallest Values of Functions.

Example 1. Break up a positive number a into three non- negative numbers so that their product should have the largest possible value.

Solution. Let the required numbers be x, y, a - x - y. Seek the maximum of the function f(x, y) = xy(a - x - y).

The function f(x, y) is to be considered inside a closed triangle (Fig. 8)

Solving the system of equations

Fig.8

we will have the unique stationary point (a/3, a/3) for the Interior of the triangle. Let us test the sufficiency conditions. We have

Consequently,

Thus, at (a/3, a/3), the function has a maximum. Since f(x, y)=0 on the border of the triangle, this maximum will be largest, i.e., the product will be largest if and the largest value is a³/27.

Note. The problem can also be solved by the methods of a conditional extremum by seeking the maximum of the function u=xyz subject to the condition x + y + z = a.

Control questions

1. Give the definition of an extreme value of a function.

2. What are the necessary conditions for an extreme value?

3. What are the sufficient conditions for an extreme value?

4. Where the function, which is differentiable in a limited closed region, attains its largest (smallest) value?