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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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Most popular questions from this chapter

Given a function of three variables, w=f(x,y,z),and a constraint equation g(x,y,z)=0,how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

In Exercises, by considering the function f(x,y)=x2ysubject to the constraint x+y=0,you will explore a situation in which the method of Lagrange multipliers does not provide an extremum of a function.

Optimize f(x,y)=x2ysubject to the constraint ax+by=0for nonzero constants a and b. Are there any nonzero values of a and b for which the method of Lagrange multipliers succeeds?

Describe the meanings of each of the following mathematical expressions:

∇f(x,y,z)

Evaluate the following limits, or explain why the limit does not exist.

lim(x,y)→(1,2)x2+y2x2-y2

Sketch the level curves f(x, y) = c of the following functions for c = −3, −2, −1, 0, 1, 2, and 3:

f(x,y)=−2yx
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