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Chapter 12: Q. 20 (page 988)

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.

A function fof two or three variables has a critical point at a point P.

Short Answer

Expert verified

A function fof two or three variables has a critical point at a point Pin the domain of a function f if Pis a stationary point of for if fis not differentiable at P.

Step by step solution

01

Step 1. Given information

A function fof two or three variables has a critical point at a point P.

02

Step 2. Defining critical point

A function fof two or three variables has a critical point at a point Pin the domain of a function f if Pis a stationary point of for if fis not differentiable at P.

When a function is not differentiable at a point, its gradient there is either zero or undefined. Points at which either of these things occur are called critical points.

Example:

(1) The function, f(x,y)=1+x2+y2has a global minimum at (0,0). It is also a critical point as it is a stationary point and gradient is zero.

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