?

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 determinantis 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

?

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 determinantis 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

has its

Constraints:

and .

, ,

, , .

The point is a critical point? The point is a.

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

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

x y

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)over the closed triangular region with vertices , and . Do it by the following questions :

First: How many critical points are in the interior of ?

- Second: fill in the coordenates of the critical point
- Calculate the value of the function
at this point :
- Is the problem finished, without any further calculations ?

- What is the value of the function on the edge (A and B included in the edge)?
- What is the value of the function on the edge (B and C included in the edge)?
- What is the value of the function on the edge (C and A included in the edge)?
- What is the value of the function on

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

- with a vertex in ,
- whose edges are parallel to the coordinate axes,
- which is in the first octant (nonnegative coordinates),
- The vertex opposite to is in the plane of equation

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

,

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.

> 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 ifThere 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

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+()

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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