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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.

User Hamady
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1 Answer

4 votes

Answer:

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.

User Gathole
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8.1k points
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