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Calc 3 11.7

 

<== Calculus 3
Number Question Answer
See the work
1. Suppose f (x, y) = xy - ax - by.

(A) How many local minimum points does f have in \mathbf{R}^{2}? (The answer is an integer).

(B) How many local maximum points does f have in \mathbf{R}^{2}?

(C) How many saddle points does f have in \mathbf{R}^{2}?

A) 0

B) 0

C) 1
The work
2. Consider the function f (x, y) = xsin(y).

(A) How many local minima does f have in \mathbf{R}^{2}?

(B) How many local maxima does f have in \mathbf{R}^{2}?

(C) How many saddle points does f have in \mathbf{R}^{2}?

A) 0

B) 0

C) ∞
The work
3. Consider the function z = x^{7}y  - 14 x^{6}  + 128
y. Find and classify all critical points of the function.

The critical point with the smallest x-coordinate is
(, ) Classification:

Critical Point = (-2, -6)
Classification: saddle point
The work
4. Consider the function

f(x,y) = e^{-8 x - x^2  - 4 y - y^2}.
Find and classify all critical points of the function.

f_x =
f_y =
f_{xx} =
f_{xy} =
f_{yy} =


The critical point with the smallest x-coordinate is
(, ) Classification:










Critical point: ( -4, -2)
Classification: Local Maximum
The work
5. Consider the function

f(x,y) = e^{-8 x - x^2  - 4 y - y^2}.
Find and classify all critical points of the function.

f_x =
f_y =
f_{xx} =
f_{xy} =
f_{yy} =


The critical point with the smallest x-coordinate is
(, ) Classification:








-2

Critical point: ( 1, 6)
Classification: Local Maximum
The work
6. Consider the function

f(x,y) = e^{-2 x} \cos (5 y)
Find and classify all critical points of the function.

f_x =
f_y =
f_{xx} =
f_{xy} =
f_{yy} =


The critical point with the smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:










The work
7. Consider the function

f(x,y) = (12 x - x^2)(8 y - y^2)
Find and classify all critical points of the function.

f_x =
f_y =
f_{xx} =
f_{xy} =
f_{yy} =


The critical point with the smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:

The critical point with the next smallest x-coordinate is
(, ) Classification:










Critical point: ( 0, 0)
Classification: Saddle Point

Critical point: ( 0, 8)
Classification: Saddle Point

Critical point: ( 6, 4)
Classification: Local Maximum

Critical point: ( 12, 0)
Classification: Saddle Point

Critical point: ( 12, 8)
Classification: Saddle Point

The work
8. Find the absolute minimum and absolute maximum of

f(x,y) = 4 - 6 x + 10 y
on the closed triangular region with vertices (0,0), (10, 0) and (10, 13).

List the maximum/minimum values as well as the point(s) at which they occur.

Minimum value:
Occurs at (,)

Maximum value:
Occurs at (,)
Minimum value: -56 at ( 10, 0)

Maximum value: 74 at ( 10, 13)
The work
9. Find the absolute maximum and absolute minimum of the function f(x,y) = 2x^3 + y^4 on the region \lbrace (x,y) | x^2 + y^2 \le 16
brace

Absolute minimum value:
attained at (, ) .

Absolute maximum value:
attained at (, ), (, ).
Absolute minimum value: -128
attained at ( -4, 0) .

Absolute maximum value: 256
attained at ( 0, -4), ( 0, 4).
The work
10. Find the absolute maximum and minimum of the function f(x,y) = y
\sqrt x - y^2 - x + 3y on the domain 0 \le x \le 9,\ 0 \le y \le
7.

Absolute minimum value:
attained at (, ).

Absolute maximum value:
attained at (, ).
Absolute minimum value: -28
attained at ( 0, 7).

Absolute maximum value: 3
attained at ( 1, 2).
The work
11. Find the coordinates of the point (x, y, z) on the plane z = 2 x + 3 y + 1 which is closest to the origin.
x =
y =
z =
x = -0.143
y = -0.2143
z = 0.07143
The work
12. Find the point(s) on the surface z^2 = xy + 1 which are closest to the point (10, 11, 0).

(6,8,-7), (6,8,7) The work
13. Find three positive real numbers whose sum is 73 and whose product is a maximum.
, ,
73/3, 73/3, 73/3 The work
14. Find the dimensions of the rectangular box having the largest volume and surface area 122 square units.
, ,.
4.509, 4.509, 4.509 The work
15. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes, and one vertex in the plane x +2 y + 3 z = 6. Largest volume is 4/3 The work