Question

Describe the level surfaces of the function f(x y z)=x+3y+5z

Answer 1

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Answer 2

The level surfaces of the function f(x, y, z) = x+3y+5z gives us a family of parallel planes.

Given:

The function f(x,y,z)=x+3y+5z

To Find:

Nature of the level surfaces of the function.

Solution:

A surface S in R³ is called a level surface of f(x,y,z) if the value of f at every point is some constant C.

The given function is f(x,y,z) = x+3y+5z

Take f(x,y,z) = k, a constant value.

⇒ x+ 3y+5z = k   ...........................(I)

We know that plane in three-dimensional space has the equation

ax + by + cz + d = 0.

Hence, equation (I) is the equation of a plane.

So for varying values of k, we will obtain different parallel planes.

∴ The level surfaces of the function f(x, y, z) = x+3y+5z gives us a family of parallel planes.

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