   Chapter 14.4, Problem 8E Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Solutions

Chapter
Section Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 1-16, find each function’s relative maxima, relative minima, and saddle points, if they exist. z = 4 x 2 + y 2 + 4 x + 1

To determine

To calculate: The relative maxima, relative minima, and saddle points of z=4x2+y2+4x+1, if they exist.

Explanation

Given Information:

The provided function is z=4x2+y2+4x+1.

Formula used:

Differentiate z=f(x,y) with respect to x by holding y constant as:

x[f(x,y)]=fx(x,y)

Differentiate f(x,y) with respect to y by holding x constant as:

y[f(x,y)]=fy(x,y)

The every differentiable function f(x,y),

2yxf(x,y)=2xyf(x,y)

The following procedure are used to calculate relative maximum and saddle point of given function f(x,y).

Step-1: Calculate first derivative of function f(x,y) with respect to x and y.

Step-2 Calculate second derivative of function f(x,y) with respect to x and y.

Step-3: Equate first derivative of function f(x,y) with respect to x to zero.

Step-4: Equate first derivative of function f(x,y) with respect to y to zero.

Step-5: Calculate critical point of function f(x,y) that is (a,b).

Step-6: Now test function f(x,y) at critical point (a,b) for extrema and saddle point that is describe in following table.

 Critical point ∂2∂2xf(x,y) ∂2∂x2f(x,y)∂2∂y2f(x,y)−[∂2∂y∂xf(x,y)]2 Conclusion (a,b) ∂2∂2xf(a,b)>0 ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2>0 The function f(x,y) has relative minimum at point (a,b) (a,b) ∂2∂2xf(a,b)<0 ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2>0 The function f(x,y) has relative maximum at point (a,b) (a,b) ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2<0 The function f(x,y) has saddle point (a,b,f(a,b)) (a,b) ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2=0 The test gives no information.

The simple power rule of derivative:

ddx(xn)=nxn1

The constant rule of derivative:

ddx(c)=0

The derivative of constant coefficient:

ddx(cf(x))=cf(x)

Calculation:

Consider the function, z=4x2+y2+4x+1.

Differentiate z=f(x,y) with respect to x by holding y constant,

zx=x(4x2+y2+4x+1)=4(2x)+4=8x+4

Equate the first derivative of function z=4x2+y2+4x+1 with respect to x to zero

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

In Exercises 5-8, solve for x. 3x2x224x=0

Calculus: An Applied Approach (MindTap Course List)

In Exercises 516, evaluate the given quantity. log416

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the point of intersection of the two straight lines having the equations y =34x + 6 and 3x 2y + 3 = 0.

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Express the function in the form f g. 43. F(x) = (2x + x2)4

Single Variable Calculus: Early Transcendentals

True or False: converges absolutely.

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: If converges absolutely, then it converges conditionally.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 