Question 1086673
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<pre>
These two equalities 

x + y + z = 7        (1)     and
xy + yz + zx = 11    (2)

imply

{{{(x + y + z)^2}}} = {{{7^2}}},

{{{x^2 + y^2 + z^2 + 2xy + 2yz + 2zx}}} = {{{7^2}}},

{{{x^2 + y^2 + z^2}}} + 2*11 = 49,

{{{x^2 + y^2 + z^2}}} = 49 - 22 = 27.


I am repeating it again:  <U>(1) and (2) imply</U>

{{{x^2 + y^2 + z^2}}} = 27.    (3)


Equation (3) defines the sphere in 3D space.

Equation (1) represents the plane in 3D.


    Therefore, I can state that equations (1) and (2) define the section of the sphere (3) by the plane (1).


    Having this geometric interpretation, we can <U>turn ON</U> our geometric intuition.


It becomes clear that the maximum and the minimum of "z" are achieved at the plane x = y.


The section of the sphere {{{x^2 + y^2 + z^2}}} = 27 by the plane x = y is the circle of the radius of {{{sqrt(27)}}} centered at the origin 
of the coordinate system (every section of a sphere by a plane through it center is the great circle of the sphere).


    Therefore, the maximal value of z is the intersection point of this circle section at the plane x = y and the straight line L
    which is the intersection of the plane x+y+z = 7  and the plane x = y.   (Obviously, the same is true for the minimal value of z, too).


Let us go to this plane x = y and introduce the axis and the coordinate "u" in this plane orthogonal to z-axis.

In this plane the equation of the circle is

{{{u^2 + z^2}}} = 27      (4)

and the equation of the straight line L is

z = {{{7 - sqrt(2)*u}}}      (5)


Thus to find {{{Z[max]}}} we need to solve the system of two equations (4), (5).

So make the substitution, simplify . . . and you will get . . . (I just did everything for you . . . )


{{{u[1,2]}}} = {{{(14*sqrt(2) +- 8*sqrt(2))/6}}}.


The smaller root for u is  {{{u[1]}}} = {{{sqrt(2)}}}, and the corresponding value of  {{{Z[max]}}}  is {{{Z[max]}}} = {{{7 - sqrt(2)*sqrt(2)}}} = 7 - 2 = 5.


The larger root for u is  {{{u[2]}}} = {{{(22*sqrt(2))/6}}},  and the corresponding value of  {{{Z[min]}}}  is  {{{Z[min]}}}= {{{7 - sqrt(2)*(22*sqrt(2))/6}}} = {{{7 - 44/6}}} = {{{-2/6}}} = {{{-1/3}}}.
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<U>Answer</U>.  &nbsp;&nbsp;{{{Z[max]}}} = 5.  &nbsp;&nbsp;{{{Z[min]}}} = {{{-1/3}}}.



Solved.