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Applied Calculus

7th Edition
Waner + 1 other
Publisher: Cengage Learning,
ISBN: 9781337291248

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BuyFindarrow_forward

Applied Calculus

7th Edition
Waner + 1 other
Publisher: Cengage Learning,
ISBN: 9781337291248
Chapter 8, Problem 31RE
Textbook Problem
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In Exercises 31-34, use Lagrange multipliers to solve the given optimization problem.

Find the minimum value of f ( x , y ) = x 2 + y 2 subject to x y = 2 . Also find the corresponding point(s) ( x , y ) .

To determine

To calculate: The minimum value of f(x,y)=x2+y2 subject to xy=2.

Explanation of Solution

Given Information:

The provided function is f(x,y)=x2+y2 subject to xy=2.

Formula used:

Steps to find the relative extrema of a function f(x,y,) subject to the constraint g(x,y,)=0,

Step 1: Form a Lagrangian function,

L(x,y,)=f(x,y,)λg(x,y,)

Where, λ is the Lagrange multiplier.

Step 2: Now, set all the partial derivatives of L(x,y,) equal to zero and find the solution of the system with the provided constraint equation g(x,y,)=0 to find the x,y, and λ.

Now, the relative extrema occur at the critical points of L(x,y,).

Calculation:

Start by rewriting the problem with the constraint in the form g(x,y)=0.

Minimize f(x,y)=x2+y2 subject to xy=2.

Here,

g(x,y)=xy2

And, the Lagrangian function is,

L(x,y)=f(x,y)λg(x,y)=x2+y2λ(xy2)

The system of equations needed to be solved are,

Lx=0x{x2+y2λ(xy

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Chapter 8 Solutions

Applied Calculus
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Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 11-18, classify each function as...Ch. 8.1 - In Exercises 19 and 20, use the given tabular...Ch. 8.1 - In Exercises 19 and 20, use the given tabular...Ch. 8.1 - In Exercises 21 and 22, use a spreadsheet or some...Ch. 8.1 - In Exercises 21 and 22, use a spreadsheet or some...Ch. 8.1 - The following statistical table lists some values...Ch. 8.1 - The formula for body mass index M(w,h), if w is...Ch. 8.1 - In Exercises 25-28, use either a graphing...Ch. 8.1 - In Exercises 25-28, use either a graphing...Ch. 8.1 - In Exercises 25-28, use either a graphing...Ch. 8.1 - In Exercises 25-28, use either a graphing...Ch. 8.1 - Brand Zs annual sale are affected by the ales of...Ch. 8.1 - Brand Zs annual sale are affected by the sales of...Ch. 8.1 - Sketch the cube with vertices...Ch. 8.1 - Sketch the cube with vertices (1,1,1),(1,1,1),...Ch. 8.1 - Sketch the pyramid with vertices (1,1,0),(1,1,0),...Ch. 8.1 - Sketch the solid with vertices (1,1,0),(1,1,0),...Ch. 8.1 - In Exercises 35-40, sketch the given plane. z=2Ch. 8.1 - In Exercises 35-40, sketch the given plane. z=4Ch. 8.1 - In Exercises 35-40, sketch the given plane. y=2Ch. 8.1 - In Exercises 35-40, sketch the given plane. y=3Ch. 8.1 - In Exercises 35-40, sketch the given plane. x=3Ch. 8.1 - In Exercises 35-40, sketch the given plane. x=2Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 41-48, match the given equation with...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - In Exercises 4954, sketch the level curves...Ch. 8.1 - Exercises 55-58 refer to the following plot of...Ch. 8.1 - Exercises 55-58 refer to the following plot of...Ch. 8.1 - Exercises 55-58 refer to the following plot of...Ch. 8.1 - Exercises 55-58 refer to the following plot of...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - In Exercises 59-74, sketch the graph of the...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - For Exercises 65-74, we suggest the use of...Ch. 8.1 - Cost Your weekly cost (in dollars) to manufacture...Ch. 8.1 - Cost Your weekly cost (in dollars) to manufacture...Ch. 8.1 - Cost Your sales of online video and audio clips...Ch. 8.1 - Cost Your Cabaret nightspot Jazz on Jupiter has...Ch. 8.1 - Scientific Research In each year from 1983 to 2003...Ch. 8.1 - Scientific Research The number z of research...Ch. 8.1 - Market Share in the I990s: Chrysler, Ford, General...Ch. 8.1 - Market Share in the I990s: Kellogg, General Mills,...Ch. 8.1 - Prison Population The number of prisoners in...Ch. 8.1 - Prison Population The number of prisoners in state...Ch. 8.1 - Marginal Cost (Interaction Model) Your weekly cost...Ch. 8.1 - Marginal Cost (Interaction Model) Repeat Exercise...Ch. 8.1 - Online Revenue Your major online bookstore is in...Ch. 8.1 - Online Revenue Repeat Exercise 87 using the...Ch. 8.1 - Sales: Walmart, Target The following table shows...Ch. 8.1 - Sales: Nintendo, Nokia The following table shows...Ch. 8.1 - Utility Suppose your newspaper is trying to decide...Ch. 8.1 - Housing Costs10? 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Explain why the...Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1-16, compute the given integral....Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 1724, find Rf(x,y)dxdy, where R is...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 25-30, find the average value of the...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - In Exercises 31-36, sketch the region over which...Ch. 8.5 - Find the volume under the graph of z=1x2 over the...Ch. 8.5 - Find the volume under the graph of z=1x2 over the...Ch. 8.5 - Find the volume of the tetrahedron shown in the...Ch. 8.5 - Find the volume of the tetrahedron with corners at...Ch. 8.5 - Productivity A productivity model at the Handy...Ch. 8.5 - Productivity Repeat Exercise 41 using the...Ch. 8.5 - Revenue Your latest game app is expected to sell...Ch. 8.5 - Revenue Your latest solid-state drive is expected...Ch. 8.5 - Revenue Your self-published novel has demand...Ch. 8.5 - Revenue Your self-published book of poetry has...Ch. 8.5 - Population Density The town of West Podunk is...Ch. 8.5 - Population Density The town of East Podunk is...Ch. 8.5 - Temperature The temperature at the point (x,y) on...Ch. 8.5 - Temperature The temperature at the point (x,y) on...Ch. 8.5 - Explain how double integrals can be used to...Ch. 8.5 - Explain how double integrals can be used to...Ch. 8.5 - Complete the following: The first step in...Ch. 8.5 - If the units of f(x,y) are zonars per square meter...Ch. 8.5 - If the units of abr(x)s(x)f(x,y)dydx are...Ch. 8.5 - Complete the following: If the region R is bounded...Ch. 8.5 - Show that if a, b, c, and d are constant, then...Ch. 8.5 - Refer to Exercise 57. If a, b, c, and d are...Ch. 8 - Let f(x,y,z)=xy+xz+x2y. Evaluate...Ch. 8 - Let g(x,y,z)=xy(x+yz)+x2. Evaluate...Ch. 8 - Let f(x,y,z)=2.720.32x3.21y+12.5z. Complete the...Ch. 8 - Let g(x,y,z)=2.16x+11y1.53z+31.4. Complete the...Ch. 8 - In Exercises 5 and 6, complete the given table for...Ch. 8 - In Exercises 5 and 6, complete the given table for...Ch. 8 - Give a formula for a (single) function f with the...Ch. 8 - Let f(x,y)=x2+(y+1)2. Show that f(y,x)=f(x+1,y1).Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 9-14, sketch the graph of the given...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 15-20, compute the partial...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 21-26, locate and classify all...Ch. 8 - In Exercises 27-30, solve the given constrained...Ch. 8 - In Exercises 27-30, solve the given constrained...Ch. 8 - In Exercises 27-30, solve the given constrained...Ch. 8 - In Exercises 27-30, solve the given constrained...Ch. 8 - In Exercises 31-34, use Lagrange multipliers to...Ch. 8 - In Exercises 31-34, use Lagrange multipliers to...Ch. 8 - In Exercises 31-34, use Lagrange multipliers to...Ch. 8 - In Exercises 31-34, use Lagrange multipliers to...Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - In Exercises 35-40, compute the given quantities....Ch. 8 - Website Traffic OHaganBooks.com has two principal...Ch. 8 - Productivity Billy-Sean OHagan is writing his PhD...Ch. 8 - Internet Advertising To increase business at...Ch. 8 - Company Retreats Their companies having recently...Ch. 8 - Internet Advertising Refer to the model in...Ch. 8 - Company Retreats Refer to the model in Exercise...Ch. 8 - Productivity The holiday season is now at its...Ch. 8 - Productivity The holiday season has now ended, and...Ch. 8 - Profit If OHaganBooks.com sells x paperback books...Ch. 8 - Cost It costs Duffin House C(x,y)=x2+2y dollars to...

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