Question

Describe the level surfaces of the following functions. f(x, y, z) = x ²− y ²− z ² .​

Answer 1

The level surfaces of the function f(x, y, z) = x² - y² - z² are hyperboloids and cones, depending on the value of the constant set for the function.

Step-by-step explanation:

The student asks about the level surfaces of the function f(x, y, z) = x² - y² - z². To describe these surfaces, we look for all points (x, y, z) such that f(x, y, z) is constant. For example, if we set f(x, y, z) = c, where c is a constant, we then solve the equation x² - y² - z² = c.

For different values of c, we have different types of surfaces:

• If c > 0, we get a two-sheet hyperboloid.
• If c = 0, the surface is a cone, because this is the point where the two sheets of the hyperboloid meet.
• If c < 0, we get a one-sheet hyperboloid.

To conclude, the level surfaces of the function are hyperboloids and cones, depending on the value of the constant c.