Question 1

Which of the following best describes a local maximum of a function $$f(x, y) at $$the point $$(a, b)$$?

Accepted Answer

It is a value $$f(a, b)$$ such that $$f(a, b) \geq f(x, y)$$ for all $$(x, y)$$ near $$(a, b).$$

Question 2

Given $$f(x, y) = y^2$ - x^2$, what type of critical point is at (0$$, 0)$$?

Accepted Answer

Saddle point

Question 3

Suppose $$f(x, y)$$ has continuous second partial derivatives and a critical point at $$(a, b). If $$the Hessian determinant $$D = f_{xx}(a, b) f_{yy}(a, b) - [f_{xy}(a, b)]^2 is $$positive and $$f_{xx}(a, b) \u003c 0$$, what does this imply?

Accepted Answer

There is a local maximum at $$(a, b).$$

Question 4

Which of the following is NOT a necessary condition for a point $$(a, b) to be a $$critical point of $$f(x, y)$$?

Accepted Answer

$$f_{xy}(a, b) = 0$$

Question 5

A hospital is optimizing the temperature and humidity in an operating room to minimize infection risk, modeled by $$R(T, H). If$$ $$R$$ has a critical point at $$(T_0$, H_0$)$$ and the Hessian determinant $$D \u003e 0$$ and $$R_{TT}(T_0$, H_0$) \u003e 0$$, what does this indicate?

Accepted Answer

$$(T_0$, H_0$) is a $$local minimum of infection risk.

Question 6

Which statement about the Extreme Value Theorem for functions of two variables is correct?

Accepted Answer

A continuous function $$f(x, y) on a $$closed, bounded set has an absolute maximum and minimum.

Question 7

For $$f(x, y) = x^2$ + y^2$ on the disk D$$ = \{(x, y) | $$x^2$$ + $$y^2$$ \leq 9\}$$, where does the absolute maximum occur?

Accepted Answer

At any point on the boundary where $$x^2$$ + $$y^2 = 9$

Question 8

A company models profit as $$P(x, y)$$, where $$x$$ and $$y$$ are quantities of two products. To find the absolute maximum profit on a closed, bounded region $$D$$, which steps should be followed?

Accepted Answer

Find values of $$P at $$critical points and on the boundary of $$D$$, then compare.

Question 9

Suppose $$f(x, y)$$ has a critical point at $$(a, b)$$ and the Hessian determinant $$D(a, b) \u003c 0. $$What does this imply?

Accepted Answer

There is a saddle point at $$(a, b).$$

Question 10

Which of the following is true about a bounded set in $$\mathbb{R}^2$$?

Accepted Answer

It is contained within some disk.

Question 11

If $$f(x, y) is $$continuous on a closed, bounded set $$D$$ and has a critical point at $$(a, b)$$ inside $$D$$, which of the following must be true?

Accepted Answer

$$f(a, b) is a $$candidate for an absolute extremum.

Question 12

A chemical reaction rate $$R(x, y)$$ depends on temperature $$x$$ and pressure $$y. If$$ $$R$$ has a local maximum at $$(x_0$, y_0$)$$, which must be true?

Accepted Answer

$$R_x(x_0$, y_0$) = 0$$ and $$R_y(x_0$, y_0$) = 0$$

Question 13

Which of the following is the correct definition of a critical point for a function $$f(x, y)$$?

Accepted Answer

A point where $$f_x = 0$$ and $$f_y = 0.$$

Question 14

For $$f(x, y) = x^2$ - 2xy + y^2$ on the rectangle 0$$ \leq x \leq 3$$, 0$$ \leq y \leq 2$$, which of the following is a correct step to find the absolute maximum and minimum?

Accepted Answer

Find critical points inside the rectangle and evaluate $$f on $$the boundary.

Question 15

If the Hessian determinant $$D(a, b) = 0 at a $$critical point $$(a, b)$$, what can be concluded?

Accepted Answer

The test is inconclusive at $$(a, b).$$

Question 16

A function $$f(x, y) is $$continuous on a closed disk $$D. $$Which of the following is guaranteed by the Extreme Value Theorem?

Accepted Answer

There exists at least one absolute maximum and one absolute minimum of $$f on$$ $$D.$$

Question 17

Which of the following is a correct application of the Second Derivative Test for functions of two variables?

Accepted Answer

If $$D \u003c 0$$, then there is a saddle point.

Question 18

A closed set in $$\mathbb{R}^2 is $$defined as:

Accepted Answer

A set that contains all its boundary points.

Question 19

If $$f(x, y) is $$continuous on a closed, bounded set $$D$$ and $$(a, b) is a $$boundary point of $$D$$, which of the following is true?

Accepted Answer

$$f(a, b)$$ can be an absolute maximum or minimum.

Section 4.7 - Maximum and Minimum Values for Two-Variable Functions

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