1
$\begingroup$

If $x,y,z$ are real numbers such that $x+y+z = 5$ and $xy+yz+zx = 3$, what is the difference between the largest and smallest possible values of any of these numbers?

What is wrong with this approach?

We have that $(x+y+z)^2 = x^2+y^2+z^2+2(xy+yz+xz) = 25 \implies x^2+y^2+z^2 = 19$. Thus the points $(x,y,z)$ lie on a sphere with radius $\sqrt{19}$. Therefore the maximum value of any of these is $\sqrt{19}$ and the smallest is $-\sqrt{19}$ and the difference is $2\sqrt{19}$.

$\endgroup$
1
  • 1
    $\begingroup$ Your statement We have that $(x+y+z)^2 = x^2+y^2+z^2 = 25-2(3) = 19$. is not correct. It should be $x^2+y^2+z^2 =(x+y+z)^2 -2(xy+yz+xz)= 25-2(3) = 19$ $\endgroup$ – Ross Millikan Jan 6 '16 at 5:33
1
$\begingroup$

$(x,y,z)$ is not $\bf{three}$ points. It is a single point in three dimensional space, where $x,y,z$ are the coordinates.

Now think of a sphere of radius $\sqrt{19}$. If the $x$ coordinate is $\sqrt{19}$, then is it possible for any other coordinate to be $-\sqrt{19}$?

$\endgroup$
1
$\begingroup$

The problem with your approach is that the point $(x,y,z)$ is not only on the sphere $x^2+y^2+z^2=19$, it is also on the plane $x+y+z=5$. That is, it is on the intersection of this sphere and plane, which is a circle. You will need to find the maximum and minimum coordinates of points on this circle.

As far as I can see this problem is quite tricky to visualise geometrically. It can be solved by Lagrange multipliers (though since you tagged the question algebra-precalculus that is probably not the method you are expected to use). If my calculations are correct the maximum possible difference is $8/\sqrt3$.

$\endgroup$
0
$\begingroup$

You have shown that the points are on the sphere you say, but it does not establish they can be anywhere on the sphere. If I try to plug $x=\sqrt{19}$ into the equations, $y$ and $z$ come out complex, so $x$ cannot be that large.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.