0
$\begingroup$

Let $f(x,y,z)=e^x(xy-y^2-z^2)$ and let A be the critical point nearest to the origin while B is the critical point furthest from the origin

Find the x.y and z coordinates of A and B

What i did

For A, i first find the partial derivatives of the original function then equating it to 0

$f_{x}=0$

$f_{y}=0$

$f_{z}=0$

which gives a critical point of (0,0,0).

From here i let A be$$(x,y,z) $$ Then using distance formula i let the distance of A to the origin be

$$\sqrt{(x^2-0)+(y^2-0)+(z^2-0)}$$

and do a partial derivative of this function to find the minimum point of this function

and equating it to 0 to find (x.y.z) hence finding out the coordinates of A which

turns out to be(0,0,0) which is the correct answer.

However, when i tried to find B using the same method as above, the method above dosent

work. Could anyone explain whether could i use the method above to find B. And if it is

not possible, then how to find B? Thanks

$\endgroup$

1 Answer 1

2
$\begingroup$

There results $$ \begin{align} \partial f / \partial x &= e^x(y+xy-y^2-z^2) \\ \partial f / \partial y &= e^x(x-2y) \\ \partial f / \partial z &= -2 e^x z. \end{align} $$ Hence $z=0$, $x=2y$, and $y(y+1)=0$. You are missing the solution $y=-1$, $x=-2$ and $z=0$.

$\endgroup$
1
  • $\begingroup$ @yswong No, all the points are critical points, just take the one closest and furthest from the origin. $\endgroup$
    – 5xum
    Sep 3, 2014 at 11:35

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .