MATLAB Differential and Integral Calculus by César Pérez López

Author:César Pérez López
Language: eng
Format: epub
Publisher: Apress, Berkeley, CA

extrema(expression,{equ1,...,equn},{variable1,...,variablen},n) The variable n returns the set of values obtained by evaluating the expression (objective function) at every candidate extreme point. These values will be used to differentiate between minima and maxima.

EXERCISE 4-20

Find and classify the extreme points of the function:

f(x, y, z) = x + z subject to the constraint x 2 + y 2 + z 2 = 1.

>> maple ('readlib (extrema)');

>> maple ('extrema(x+z,{x^2+y^2+z^2=1},{x,y,z},s)')

ans =

{2 ^(1/2), - 2 ^(1/2)}

The command extrema returns the values that the objective function takes at each extreme point and the variable s contains the extreme points.

>> maple ('s ')

ans =

{{z = 1/2 * 2 ^(1/2), x = 1/2 * 2 ^(1/2), y = 0}, {x = - 1/2 * 2 ^(1/2)}}

{{z = - 1/2 * 2 ^(1/2), y = 0}}

Thus the extreme points are:

(Ö2/2, 0, Ö2/2) and (−Ö2/2, 0, −Ö2/2)

Now let’s determine the nature of these extreme points. To do this, we evaluate the objective function at these points (we already know that the results are Ö2 and −Ö2).

>> maple ('obj:=(x,y,z) - > x+z');

>> maple('obj(sqrt(2)/2,0,sqrt(2)/2)')

ans =

2^(1/2)

>> maple('obj(-sqrt(2)/2,0,-sqrt(2)/2)')

ans =

-2 ^(1/2)

Then, at the point (−Ö2 /2, 0,−Ö2 /2) the function has a minimum, and there is a maximum at the point (Ö2/2,0,Ö2/2).

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