OEF min-max2 --- Introduction ---

This module actually contains 12 exercises on the extrema of functions of two variables.

Distance from a point to a conic

What is the distance from the origin to the curve of equation

?

xrange -, + yrange -, + parallel -,0,+,0,0,1,40,grey parallel -,0,+,0,0,-1,40,grey parallel 0,-,0,+,1,0,40,grey parallel 0,-,0,+,-1,0,40,grey arrow 0,0, 0,1,10,black arrow 0,0, 1, 0 ,10,black vline 0,0, black hline 0,0, black levelcurve darkred, , levelcurve green,

This problem is interpreted as a constrained minima problem. Solve by by answering the following questions:

First:

It is intended to find the minimum of the function of two variables defined by

subject to the constraint

Second: With and we have

grad ( , )

grad ( , )

These two vectors are related to point if their determinant

is equal to . Third: Remember that grad and grad are parallel if

Give the value of the critical point for which is minumum on the curve .

A=( , )

The minimum distance of the curve at the point is therefore equal to

The exercise has several steps


Distance from a point to a curve*

What is the distance from the origin to the curve of equation

?

xrange -, + yrange -, + parallel -,0,+,0,0,1,40,grey parallel -,0,+,0,0,-1,40,grey parallel 0,-,0,+,1,0,40,grey parallel 0,-,0,+,-1,0,40,grey arrow 0,0, 0,1,10,black arrow 0,0, 1, 0 ,10,black vline 0,0, black hline 0,0, black levelcurve darkred, , levelcurve green,

This problem is interpreted as a constrained minima problem. Solve by answering the following questions:

First:

It is intended to find the minimum of the function of two variables defined by

subject to the constraint

Second: With and , we have

grad ( , )

grad ( , )

These two vectors are parallel in the point if their determinant

is equal to . Third: Remember that grad and grad are parallel if

Give the value of the critical point for which is minumum on the curve .

A=( , )

The minimum distance of the curve at the point is therefore equal to


Affine function on an ellipse

Find the point, subject to the constraints below, where the function defined by

has its

Constraints:

and .


Extremum I

Let be a function of two variables, from to of class , and a point in . Assume that:

, ,

, , .

The point is a critical point? The point is a

Extremum II

Let a function from to defined by

.

The point is a :

Critical point and level curves

Let be a function of two variables, from to of class , and a point in . Assume that in a neighbourdhood of a point , we have:

psi

where psi is a function of class such that psi =0. Which of these three designs may represent the contours of in a neighbourhood of ?


Maximum (with edges, pipe)

A pipe of trapezoidal profile is constructed by bending a plate of total width . Let be the bend width at either side and the angle that the bend portions make with the vertical (see drawing below). What are the values of and that maximize the pipe flow?

xrange -3.5, 3.5 yrange -3.5, 3.5 linewidth 3 arc 3.3, 3,3, 240,300, black lines green ,-3,1,-2,-1, 2,-1,3,1 line -3,1,3,1, red fill 0,0, skyblue transparent red xrange -4.5, 4.5 yrange -4.5, 4.5 line -2,-1,-2,1,black text black,-3,1, medium, y text black,-3.5,0, medium, x arc -2,-1,2,2, 90,130, black linewidth 3 lines green ,-4,1,-2,-1, 2,-1,4,1

Solve by answering the following steps:

First: The surface we want to maximize is defined by the function given by

on a domain defined by

leq x leq    leq y leq

Second: Calculate the critical point of which is within the domain , )

and the function value in this point : For your information, here are the level curves of xrange -0.5, yrange -0.5, parallel -1,0,,0,0,1,40,grey parallel -1,0,,0,0,-1,40,grey parallel 0,-1,0,,1,0,40,grey parallel 0,-1,0,,-1,0,40,grey levelcurve darkred, , vline 0,0, black hline 0,0, black linewidth 3 lines green, 0,0,0,pi/2, /2,pi/2, /2,0,0,0 linewidth 1 arrow 0,0, 0,1,10,black arrow 0,0, 1, 0 ,10,black disk ,, 5,blue

Third: Now compute what the maximum of is, if are taken on the edge of Fourth: So, as a conclusion, the maximun flow is obtained for a value of and a value of the angle (in radians)


Extremum on an area

We want to find the global of the function defined by

over the closed triangular region with vertices , and . Do it by the following questions :
First: How many critical points are in the interior of ? Third: We have .

xrange , yrange , fpoly grey,,, text black, , medium, A text black, , medium,B text black, , medium,C


Maximum volume with Conditions I

Calculate the volume of the largest rectangular parallelepiped Solve the problem by answering the following questions:

First: calculate the volum as a function of . Second: The domain of the function given by corresponding to this problem is a triangle

leq leq   ,   leq leq

Third: How many critical points of the function given by are inside Fourth: The critical point of inside is ( , ). Fifth: The function is zero at the edges of . As the function has a maximum, and the function values are positive inside and zero at the edges, the critical point is necessarily a maximum.

Calculate the lenghts , and of three sides corresponding to the box of maximum volume, and the volume.

>

Maximum volume with conditions II

We want to find what the maximum possible volume of a rectangular box without top, made with of cardboard, is

Solve by answering the following questions


This problem is interpreted as a constrained maxima problem.
It is intended to find the maximum of the function of three variables defined by

subject to the constraint

With and we have

grad ( , , )

grad ( , , )

These two vectors are parallel in the point if the vector product

( , , )

is equal to . The value (giving the volume) for a point, with one zero coordenate, is equal to . And the point a maximum. Thus, grad and grad are linearly dependent if

There is only one critical point for this problem such that all its coordinates are non-zero, this is the point

( , , )

and the corresponding volume is


Least squares method

We have some evidences that the two quantities and are related by an affine relationship . The experimental data give the following points

y= x +

xrange -1,+1 yrange -1,+1 hline black, 0,0 vline black, 0,0 arrow 0,0,0,1,10,black arrow 0,0,1,0,10,black plot red , ()*x+()


Distance to the tangents of a conic

Calculate the point of the conic section of equation whose tangent to is at the largest distance to the point . The most recent version


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Description: collection of exercises on the extrema of functions of two variables. interactive exercises, online calculators and plotters, mathematical recreation and games

Keywords: interactive mathematics, interactive math, server side interactivity, analysis, maximum, minimum, extremum, point critique, maxima, minima