1. ## Calculus [2 questions]

1. Find the point of contact between surfaces x^2+y^2+z^2=8 and xy=4 where they are tangential to each other. (xy=4 is vertical cylinder)

2. Given f(x,y,z) such that the gradient vector is zero at every (x,y,z). What do we know about f? What is the implication?

2. Hello, catcat103!

1. Find the points of contact between surfaces: .$\displaystyle \begin{array}{c}x^2+y^2+z^2\:=\:8 \\ xy\:=\:4 \end{array}$
where they are tangential to each other.

Code:
                |
| o
|
|  o
* * * o
*     |     *  (2,2)
*       |       ◊
*        |        * o
|                 o
*         |         *               o
- - * - - - - * - - - - * - - - - - - - - - -
*         |         *
|
*        |        *
*       |       *
*     |     *
* * *
|

Looking at the xy-plane we have a circle with radius $\displaystyle 2\sqrt{2}$
. . and a hyperbola through the point $\displaystyle (2,2).$
They intersect at $\displaystyle (2,2).$

The other branch of the hyperbola intersects the circle at $\displaystyle (-2,-2).$

The sphere and hyperbolic cylinder intersect at: $\displaystyle (2,2,0)\text{ and }(-2,-2,0)$

3. Thanks for your solution which is easy to understand
But is there any other way that does not require plotting of graph (as it is difficult to sketch the graph sometimes)?