Question 1

Let f(x, y, z) = xyz. Find the point(s) where f(x, y, z) may have a possible relative maximum or minimum, subject to the constraint x + 6y + 3z = 36 and where x > 0, y > 0, z > 0. Use the method of Lagrange multipliers.
Enter your answer as just (a, b, c) where a, b, c are all integers.

Accepted Answer

The answer of Let f(x, y, z) = xyz. Find...

Question 2

Find all points (x, y) where $f ( x , y ) = x ^ { 3 } - y ^ { 2 } - 3 x + y + 5$ has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b), (c, d) with a > c and where a, b, c, d are either integers or reduced fractions of form $\frac { e } { f }$ .

Accepted Answer

The answer of Find all points (x, y) where \(f...

Question 3

Let f(x, y, z) = $x ^ { 2 }$ y + $\frac { x } { z }$ . Find $\frac { \partial f } { \partial z }$ .
Enter your answer as a polynomial in x in standard form (unlabeled).

Accepted Answer

The answer of Let f(x, y, z) = \(x ^...

Question 4

The function H(x, y) = $x ^ { 4 }$ - 9 $y ^ { 2 }$ - 2 $x ^ { 2 }$ y + 20y + 4 has

Accepted Answer

The function does not have a relative maximum or minimum at any of the given points based on the second derivative test.

Question 5

A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.

Accepted Answer

The surface area of the box (material required) is given by $S = xy + 2xz + 2yz$, and the volume $V = xyz = 32$. To minimize the surface area with a given volume, it's a problem of optimization under constraint, solvable via methods like Lagrange multipliers or by intuition and algebraic manipulation. Among the given options, B) 4 inches by 4 inches by 2 inches offers the least surface area for the given volume, as it approaches a form closer to a square base, which is more efficient in terms of surface area to volume ratio than a more elongated rectangular base or a cube when the top is open.

Question 6

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4).
Enter just a reduced fraction $\frac { a } { b }$ .

Accepted Answer

The answer of Let f(x, y) = \(\sqrt { x...

Question 7

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

Accepted Answer

The answer of Find the greatest possible volume of a...

Question 8

Let f(x, y) = x $e ^ { 2 } y$ + y $\mathrm { e } ^ { 2 x }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } }$ .
Enter your answer as just an unlabeled polynomial in $\mathrm { e } ^ { 2 x }$ in standard form.

Accepted Answer

The answer of Let f(x, y) = x \(e ^...

Question 9

Let f(x, y, z) = $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ . Find $\frac { \partial f } { \partial z }$ .
Enter your answer as a polynomial in $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ in standard form (unlabeled).

Accepted Answer

The answer of Let f(x, y, z) = \(e ^...

Question 10

Find all points (x, y) where $f ( x , y ) = x ^ { 3 } - 12 y + y ^ { 2 }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

The answer of Find all points (x, y) where \(f...

Question 11

Let Q(x, y) = $x ^ { 2 }$ y + $y ^ { 3 }$ $x ^ { 4 }$ . The point (1, 0) is

Accepted Answer

The function Q(x, y) = $x^2y + y^3x^4$ at the point (1, 0) simplifies to 0, since any term with y in it becomes 0. To determine the nature of this point, we need to look at the behavior of Q around (1, 0). The absence of y in the simplified expression at (1, 0) and the presence of both x and y terms in the original function suggest that the point is not a relative extreme (maximum or minimum), as the function's value can increase or decrease in the vicinity of (1, 0) depending on the direction from which it is approached.

Question 12

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

The answer of Find all points (x, y) where \(f...

Question 13

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ - 2 $y ^ { 2 }$ + 4x - 6y + 8 has a possible relative maximum or minimum.
Enter your answer as just (a, b) where a, b are either integers or reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Accepted Answer

The answer of Find all points (x, y) where f(x,...

Question 14

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Accepted Answer

To determine if the given point is a relative maximum, relative minimum, or not a relative extreme point, we can use the second partial derivative test. First, we find the first partial derivatives of $f(x, y)$:$$f_x = 2x - y$$$$f_y = -x + 2y + 2$$Setting these equal to zero to find critical points:$$2x - y = 0$$$$-x + 2y + 2 = 0$$Solving this system of equations, we find $x = -\frac{2}{3}$ and $y = -\frac{4}{3}$, which matches the given point, indicating it's a critical point.Next, we find the second partial derivatives:$$f_{xx} = 2$$$$f_{yy} = 2$$$$f_{xy} = f_{yx} = -1$$Then, we calculate the determinant of the Hessian matrix at the critical point:$$D = f_{xx}f_{yy} - (f_{xy})^2 = (2)(2) - (-1)^2 = 4 - 1 = 3$$Since $D > 0$ and $f_{xx} > 0$, the point is a relative minimum.

Question 15

Find all points (x, y) where f(x, y) = 2 $x ^ { 2 }$ + 2 $y ^ { 3 }$ - x - 6y + 14 has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b), (c, d) with b > d and where a, b, c, d are either integers or reduced fractions of form $\frac { e } { f }$ .

Accepted Answer

The answer of Find all points (x, y) where f(x,...

Question 16

Let f(x, y) = $x ^ { 4 }$ - $y ^ { 2 }$ + $3 x ^ { 2 }$ + 2y - 7. The first partial derivatives of f(x, y) are zero at the points $( 0,1 )$ and $( - 1,1 )$ Use the second derivative test to determine the nature of f(x, y) at each of these points.

Accepted Answer

The answer of Let f(x, y) = \(x ^ {...

Question 17

A tennis racket manufacturer produces two types of rackets, standard and competition. The weekly revenue function, in dollars, for x standard rackets and y competition rackets is given by R(x, y) = 54x + 2xy + 398y - 2 $x ^ { 2 }$ - 9 $y ^ { 2 }$ i) How many of each type of racket must be produced each week to maximize revenue?
Ii) What is the maximum weekly revenue?

Accepted Answer

The maximum revenue is achieved by producing 26 standard rackets and 25 competition rackets, resulting in a maximum weekly revenue of $5677.

Question 18

Find all points (x, y) where $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 } - 3 x + 2$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

DTo find the critical points of $f(x, y)$, we first need to compute the first partial derivatives of $f$ with respect to $x$ and $y$, and then set them equal to zero.$$f_x = 2x + y - 3$$$$f_y = x + 2y$$Setting $f_x = 0$ and $f_y = 0$ gives us a system of equations:$$2x + y - 3 = 0$$$$x + 2y = 0$$Solving this system, we find $x = 2$ and $y = -1$, so one critical point is $(2, -1)$.Next, we compute the second partial derivatives:$$f_{xx} = 2$$$$f_{yy} = 2$$$$f_{xy} = f_{yx} = 1$$The second-derivative test for a function of two variables is given by $D = f_{xx}f_{yy} - (f_{xy})^2$. At $(2, -1)$, we find:$$D = (2)(2) - (1)^2 = 4 - 1 = 3$$Since $D > 0$ and $f_{xx} > 0$, the function $f$ has a relative minimum at $(2, -1)$.For the point $(-2, 1)$, it seems there was a misunderstanding in the calculation as this point does not satisfy the system of equations derived from setting the first partial derivatives to zero. Therefore, only the point $(2, -1)$ is correctly identified and analyzed according to the given function and its derivatives. The correct interpretation is that $(2, -1)$ is a relative minimum, making option B correct. The mention of $(-2, 1)$ seems to be a mistake as it does not result from solving the system of equations for the critical points of $f(x, y)$. Thus, only option B is correct based on the provided analysis.

Question 19

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Accepted Answer

To determine the points where $f(x, y) = e^{(x^2 + y^2)}$ has a possible relative maximum or minimum, we first find the critical points by taking the partial derivatives of $f$ with respect to $x$ and $y$, setting them equal to zero, and solving for $x$ and $y$.The first partial derivative with respect to $x$ is $f_x = 2xe^{(x^2 + y^2)}$, and with respect to $y$ is $f_y = 2ye^{(x^2 + y^2)}$. Setting $f_x = 0$ and $f_y = 0$ gives $x = 0$ and $y = 0$, so the only critical point is $(0, 0)$.To determine the nature of this critical point, we use the second derivative test for functions of two variables, which involves computing the second partial derivatives $f_{xx}$, $f_{yy}$, and the mixed partial derivative $f_{xy}$, and then evaluating the determinant of the Hessian matrix $D = f_{xx}f_{yy} - (f_{xy})^2$ at the critical point.The second partial derivatives are $f_{xx} = (2 + 4x^2)e^{(x^2 + y^2)}$, $f_{yy} = (2 + 4y^2)e^{(x^2 + y^2)}$, and $f_{xy} = 4xye^{(x^2 + y^2)}$. At the point $(0, 0)$, these derivatives simplify to $f_{xx} = 2$, $f_{yy} = 2$, and $f_{xy} = 0$. Thus, $D = (2)(2) - (0)^2 = 4$, which is positive.Since $D > 0$ and $f_{xx} > 0$ at $(0, 0)$, the function $f(x, y)$ has a relative minimum at this point.

Question 20

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ + xy + $y ^ { 2 }$ - x - y + 2 has a possible relative maximum or minimum.
Enter your answer exactly as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers.

Accepted Answer

The answer of Find all points (x, y) where f(x,...

Quiz 7: Functions of Several Variables

Let f(x, y, z) = xyz. Find the point(s) where f(x, y, z) may have a possible relative maximum or minimum, subject to the constraint x + 6y + 3z = 36 and where x > 0, y > 0, z > 0. Use the method of Lagrange multipliers. Enter your answer as just (a, b, c) where a, b, c are all integers.

Let f(x, y, z) = x2x ^ { 2 }x2 y + xz\frac { x } { z }zx​ . Find ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ . Enter your answer as a polynomial in x in standard form (unlabeled).

The function H(x, y) = x4x ^ { 4 }x4 - 9 y2y ^ { 2 }y2 - 2 x2x ^ { 2 }x2 y + 20y + 4 has

A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.

Let f(x, y) = x2+y2\sqrt { x ^ { 2 } + y ^ { 2 } }x2+y2​ . Compute ∂f∂x\frac { \partial f } { \partial x }∂x∂f​ at (3, 4). Enter just a reduced fraction ab\frac { a } { b }ba​ .

Find the greatest possible volume of a rectangular box that has length plus girth equal to 60 inches. Enter your answer as a single integer (no units).

Let f(x, y) = x e2ye ^ { 2 } ye2y + y e2x\mathrm { e } ^ { 2 x }e2x . Compute ∂2f∂x2\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } }∂x2∂2f​ . Enter your answer as just an unlabeled polynomial in e2x\mathrm { e } ^ { 2 x }e2x in standard form.

Let f(x, y, z) = ex2+y2+z2e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }ex2+y2+z2 . Find ∂f∂z\frac { \partial f } { \partial z }∂z∂f​ . Enter your answer as a polynomial in ex2+y2+z2e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }ex2+y2+z2 in standard form (unlabeled).

Find all points (x, y) where f(x,y) =x3−12y+y2f ( x , y ) = x ^ { 3 } - 12 y + y ^ { 2 }f(x,y) =x3−12y+y2 has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Let Q(x, y) = x2x ^ { 2 }x2 y + y3y ^ { 3 }y3 x4x ^ { 4 }x4 . The point (1, 0) is

Find all points (x, y) where f(x,y) =1x+xy−1yf ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }f(x,y) =x1​+xy−y1​ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x, y) = x2x ^ { 2 }x2 - 2 y2y ^ { 2 }y2 + 4x - 6y + 8 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are either integers or reduced fractions of form cd\frac { \mathrm { c } } { \mathrm { d } }dc​ .

Let f(x, y) = x2x ^ { 2 }x2 - xy + y2y ^ { 2 }y2 + 2y - 4. The point −23,−43- \frac { 2 } { 3 } , - \frac { 4 } { 3 }−32​,−34​ is a

Find all points (x, y) where f(x, y) = 2 x2x ^ { 2 }x2 + 2 y3y ^ { 3 }y3 - x - 6y + 14 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b), (c, d) with b > d and where a, b, c, d are either integers or reduced fractions of form ef\frac { e } { f }fe​ .

Let f(x, y) = x4x ^ { 4 }x4 - y2y ^ { 2 }y2 + 3x23 x ^ { 2 }3x2 + 2y - 7. The first partial derivatives of f(x, y) are zero at the points (0,1) ( 0,1 ) (0,1) and (−1,1) ( - 1,1 ) (−1,1) Use the second derivative test to determine the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x,y) =x2+xy+y2−3x+2f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 } - 3 x + 2f(x,y) =x2+xy+y2−3x+2 has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x,y) =e(x2+y2) f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }f(x,y) =e(x2+y2) has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x, y) = x2x ^ { 2 }x2 + xy + y2y ^ { 2 }y2 - x - y + 2 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b) where a, b are reduced fractions of form cd\frac { \mathrm { c } } { \mathrm { d } }dc​ or integers.

Let f(x, y, z) = $x ^ { 2 }$ y + $\frac { x } { z }$ . Find $\frac { \partial f } { \partial z }$ . Enter your answer as a polynomial in x in standard form (unlabeled).

The function H(x, y) = $x ^ { 4 }$ - 9 $y ^ { 2 }$ - 2 $x ^ { 2 }$ y + 20y + 4 has

Let f(x, y) = $\sqrt { x ^ { 2 } + y ^ { 2 } }$ . Compute $\frac { \partial f } { \partial x }$ at (3, 4). Enter just a reduced fraction $\frac { a } { b }$ .

Let f(x, y) = x $e ^ { 2 } y$ + y $\mathrm { e } ^ { 2 x }$ . Compute $\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } }$ . Enter your answer as just an unlabeled polynomial in $\mathrm { e } ^ { 2 x }$ in standard form.

Let f(x, y, z) = $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ . Find $\frac { \partial f } { \partial z }$ . Enter your answer as a polynomial in $e ^ { x ^ { 2 } } + y ^ { 2 } + z ^ { 2 }$ in standard form (unlabeled).

Find all points (x, y) where $f ( x , y ) = x ^ { 3 } - 12 y + y ^ { 2 }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Let Q(x, y) = $x ^ { 2 }$ y + $y ^ { 3 }$ $x ^ { 4 }$ . The point (1, 0) is

Find all points (x, y) where $f ( x , y ) = \frac { 1 } { x } + x y - \frac { 1 } { y }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ - 2 $y ^ { 2 }$ + 4x - 6y + 8 has a possible relative maximum or minimum. Enter your answer as just (a, b) where a, b are either integers or reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ .

Let f(x, y) = $x ^ { 2 }$ - xy + $y ^ { 2 }$ + 2y - 4. The point $- \frac { 2 } { 3 } , - \frac { 4 } { 3 }$ is a

Find all points (x, y) where f(x, y) = 2 $x ^ { 2 }$ + 2 $y ^ { 3 }$ - x - 6y + 14 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b), (c, d) with b > d and where a, b, c, d are either integers or reduced fractions of form $\frac { e } { f }$ .

Let f(x, y) = $x ^ { 4 }$ - $y ^ { 2 }$ + $3 x ^ { 2 }$ + 2y - 7. The first partial derivatives of f(x, y) are zero at the points $( 0,1 )$ and $( - 1,1 )$ Use the second derivative test to determine the nature of f(x, y) at each of these points.

Find all points (x, y) where $f ( x , y ) = x ^ { 2 } + x y + y ^ { 2 } - 3 x + 2$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where $f ( x , y ) = e ^ { \left( x ^ { 2 } + y ^ { 2 } \right) }$ has a possible relative maximum or minimum. Use the second-derivative test to determine, if possible, the nature of f(x, y) at each of these points.

Find all points (x, y) where f(x, y) = $x ^ { 2 }$ + xy + $y ^ { 2 }$ - x - y + 2 has a possible relative maximum or minimum. Enter your answer exactly as just (a, b) where a, b are reduced fractions of form $\frac { \mathrm { c } } { \mathrm { d } }$ or integers.