Question

Suppose that f is a differentiable function of one variable. Show that all the tangent Planes to the the surface z = xf (y / x) intersect in a common point.

Answer 1

If
`P_0 (x_0,y_0,z_0)` is a point on the surface, then the cartesian equation of the tangent plane at
`P_0 (x_0,y_0,z_0)` is

`(\ast)z = z_0 + (\partial z)/( \partial x)(x_0,y_0)\cdot (x -x_0) + (\partial z)/(\partial y) (x_0, y_0) (y -y_0)`,

where
`z_0 = x_0 f \left ( (y_0)/(x_0)\right )`.

Given that

`(\partial z)/(\partial x) (x_0 , y_0) = f \left( (y_0)/(x_0)\right ) - (y_0)/(x_0) \cdot (\partial f)/(\partial x)(x_0,y_0) \ , \ (\partial z)/(\partial y) (x_0 , y_0)=(\partial f)/(\partial y) (x_0,y_0)`, then

`(\ast)` becomes

`(\ast \ast) z=x_0 f \left ( (y_0)/(x_0)\right ) + f \left( (y_0)/(x_0)\right ) - (y_0)/(x_0)\cdot (\partial f)/(\partial x) (x_0,y_0)\cdot (x -x_0)+(\partial f)/(\partial y) (x_0,y_0)\cdot (y -y_0)`.

Finally, replacing
`(x,y,z)=(0,0,0)` in
`(\ast \ast)` you have that the equality is true for all
`P_0`. This means that
`O(0,0,0)`

belongs to all tangent planes and therefore, the result follows.