# Differentiability for a Function of Two or More Variables

• Jun 14th 2011, 04:29 PM
katchat64
Differentiability for a Function of Two or More Variables
STATEMENT: Find all points (x,y) at which the tangent plane to the graph of z = x^2 - 6x + 2y^2 -10y + 2xy is horizontal.

ANSWER: The answer in the back of the book is (1,2), but I have no idea how you get that answer.

ATTEMPT: I think to start the problem you need to find the gradient vector and then use the point (0,0,1) because it's the unit vector in the z direction where it's perpendicular to the horizontal plane. That's as far as I gotten and I do not know where to go from here or if the way I started the problem is even right.

(DEL)(which is an upside down triangle) f (x,y) = (2x - 6 +2y)i + (4y - 10 +2x)j
• Jun 14th 2011, 04:56 PM
Also sprach Zarathustra
Re: Differentiability for a Function of Two or More Variables
Solve system of the two equations:

2x - 6 +2y=0
4y - 10 +2x=0

• Jun 14th 2011, 05:13 PM
katchat64
Re: Differentiability for a Function of Two or More Variables
Thank you I got it from here.
• Jun 15th 2011, 01:46 AM
HallsofIvy
Re: Differentiability for a Function of Two or More Variables
Quote:

Originally Posted by katchat64
STATEMENT: Find all points (x,y) at which the tangent plane to the graph of z = x^2 - 6x + 2y^2 -10y + 2xy is horizontal.

ANSWER: The answer in the back of the book is (1,2), but I have no idea how you get that answer.

ATTEMPT: I think to start the problem you need to find the gradient vector and then use the point (0,0,1) because it's the unit vector in the z direction where it's perpendicular to the horizontal plane. That's as far as I gotten and I do not know where to go from here or if the way I started the problem is even right.

(DEL)(which is an upside down triangle) f (x,y) = (2x - 6 +2y)i + (4y - 10 +2x)j

The LaTeX command for "del" is \nabla: $\nabla f(x,y)= (2x- 6+ 2y)i+ (4y- 10+ 2x)j$

But what I would do is define $F(x,y,z)= x^2 - 6x + 2y^2 -10y + 2xy- z$ so that $\nabla F(x, y, z)= (2x- 6+ 2y)i+ (4y- 10+ 2x)j- k$ and then argue that the tangent plane will be parallel to the xy-plane if and only if that vector is in the z-direction. That is, that 2x- 6+ 2y= 0 and 4y- 10+ 2x= 0 as Also sprach Zarathustra said.