Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes.

Question

Horizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes.

1. x² + 2y² + z² - 2x - 2z - 2 = 0

2. x² + y² - z² - 2x + 2y + 3 = 0

Answer 1

1. F(x, y, z) = x² + 2y² + z² - 2x - 2z - 2 = 0; F_x = 2x - 2 = 0; x = 1; F_y = 4y = 0; y = 0; F(1,0, z) = z² - 2z -

3 = 0; z = 3 and z = - 1 planes horizontal and points (1, 0, 3) an (1, 0, - 1)

2.F(x, y, z) = x² + y² - z² - 2x + 2y + 3 = 0; F_x = 2x - 2 = 0; x = 1; F_y = 2y + 2 = 0; y = - 1; F(1, - 1, z) = - z² + 1 = 0; z = 1 or z = -1. Horizontal planes: z =1 and z = -1 and points (1, - 1, 1) and (1, - 1, - 1)