Find the equation of the plane tangent to the surface at a given point

Question

I am given the equation of a surface: $$x^3+y^3+z^3-3xyz =0$$ And I need to find the equation of the plane tangent to this surface at $(1,1,1).$
At first, this task did not look easy for me as we are not given an explicit equation of a surface, but I tried using implicit differentiation assuming that $z$ depends on $x$ and $y$, therefore:

1. Differentiate with respect to $x$ $$3x^2+3z^2 \frac {\partial z}{\partial x} - 3yz - 3xy \frac{\partial z}{\partial x} = 0$$ and I use this equation to solve for $\frac{\partial z}{\partial x}$

2. Differentiate with respect to $y$: $$3y^2 + 3z^2 \frac{\partial z}{\partial y} - 3xz - 3xy \frac{\partial z}{\partial y} = 0 $$ And I solve for $\frac{\partial z}{\partial y}$

Now, I have both partial derivatives which I can evaluate at $(1, 1, 1)$ to obtain the equation of the plane.
Is it the correct way to solve this problem?

Answer 1

The surface can be considered a level surface of:

$$ F(x,y,z) = x^3+y^3+z^3-3xyz $$

The gradient is always orthogonal to the level surface, and thus will serve to define the normal to the tangent plane. When the gradient is zero, this doesn't apply.

The gradient of $F$ is:

$$ \nabla F = (F_x,F_y,F_z) = \vec n = ( 3x^2 - 3yz, 3y^2 -3xz, 3z^2 - 3xy ) $$

Evaluated at the point (1,1,1) the gradient is the zero vector:

$$ \nabla F(1,1,1) = (0,0,0) $$

This is a trick question, there is no tangent plane at that point. Think of the two dimensional analog with a contour plot (level curves instead of a level surface). At any given level curve, I can find the tangent line. But at a peak, which is a point on the contour map, the idea of a tangent line is undefinable.

Ced

Answer 2

The so called surface $f(x,y,z) = x^3+y^3+z^3-3 x y z= 0$ is the product of a plane and a null radius cylinder or

$$ f(x,y,z) = (x+y+z)(x^2+y^2+z^2-x y- y z - x z) = 0 $$

The plane doesn't contain the point $(1,1,1)$ and the cylinder contains it but it has null radius so it is reduced to a line (generatrix) and a line has not unique normal vector.