(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

S is the surface with equation [tex] z = x^2 +2xy+2y[/tex]a) Find an equation for the tangent plane to S at the point (1,2,9).

b) At what points on S, in any, does S have a horizontal tangent plane?

3. The attempt at a solution

[tex]F(x,y,z): z = x^2 +2xy+2y[/tex]

[tex]F_x = 2x + 2y[/tex]

[tex] F_y = 2x + 2[/tex]

Evaluated at (1,2) gives answers 6 and 4, respectively. My equation for a plane is:

[tex]z-9=6(x-1) + 4(y-1)[/tex].

I think any horizontal plane should have normal vector <0,0,k>, where k is some scalar. I'm pretty sure that S has no such normal vector. But if

[tex]F(x,y,z): 0 = x^2 +2xy+2y - z[/tex]

then

[tex]grad F = <2x + 2y,2x + 2,-1>[/tex] It seems like I can let (x,y) = (-1,1) to zero the x-, y-components of the gradient. Plugging (-1,1) into the definition of z gives z = 1. This suggests to me that there is a point (-1,1,1), at which there is a horizontal tangent plane. Yet I feel pretty sure that this isn't true!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Finding Horizontal Tangent Planes on S

**Physics Forums | Science Articles, Homework Help, Discussion**