Chapter 12: Q. 8 (page 988)

Fill in the blanks to complete each of the following theorem statements:
If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a _____ of $f$ .

Short Answer

Expert verified

If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a critical pointof $f$ .

Step by step solution

Step 1. Given information

If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a _____ of $f$ .

Step 2. Filling in the blanks

If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a critical point of $f$ .

This is because a local extremum can occur only at a critical point. A point $(x_{0}, y_{0})$ in the domain of a function $f (x, y)$ is called a critical point of $f$ if $(x_{0}, y_{0})$ is a stationary point of $f$ or if $f$ is not differentiable at $(x_{0}, y_{0})$ .

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Fill in the blanks to complete each of the following theorem statements:
If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a _____ of $f$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks

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Fill in the blanks to complete each of the following theorem statements:If f(x,y)has a local extremum at x0,y0, then x0,y0is a _____ of f.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks

One App. One Place for Learning.

Most popular questions from this chapter

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Fill in the blanks to complete each of the following theorem statements:
If $f (x, y)$ has a local extremum at $(x_{0}, y_{0})$ , then $(x_{0}, y_{0})$ is a _____ of $f$ .