Question

If f(x,y,z) is differentiable everywhere, then it will attain a global maximum

and minimum when constrained to the paraboloid z=x2+y2+3.
True False

Answer 1

The statement is false because differentiability of a function does not assure the existence of a global maximum or minimum on a surface such as the paraboloid described.

Step-by-step explanation:

A function being differentiable does not guarantee that it will attain a global maximum or minimum on a given surface.

To determine if a global maximum or minimum exists, one would often use techniques from multivariable calculus such as the method of Lagrange multipliers.

This method involves finding the points where the gradient of the function is parallel to the gradient of the constraint.

In this specific example, the paraboloid is a surface that extends infinitely, and the function f(x,y,z) could potentially increase or decrease without bound, depending on its form. Therefore, the existence of global extrema is not assured just by differentiability.

Therefore, the statement "If f(x,y,z) is differentiable everywhere, then it will attain a global maximum and minimum when constrained to the paraboloid z=x2+y2+3" is false.