Explanation Given Information: The function is f ( x , y ) = x 4 + y 4 + 4 x − 32 y + 80 . Formula used: For a function, f ( x ) the partial derivative with respect to x is calculated as, the chain rule states that, ∂ ∂ x ( f ( x ) ) n = n ( f ( x ) ) n − 1 ∂ ∂ x [ f ( x ) ] Here, f ( x ) is the function of x and n is the real number. Condition of relative maximum and minimum: D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 If the function f has two variables. It has a relative minimum at ( a , b ) if D > 0 and f x x ( a , b ) > 0 . If the function f has two variables. It has a relative maximum at ( a , b ) if D > 0 and f x x ( a , b ) < 0 . Calculation: Partially differentiate the function f ( x , y ) = 3 x − 6 y − x 2 − y 2 for the values f x , f y , f x x , f y y and f x y . Consider the function, f ( x , y ) = 3 x − 6 y − x 2 − y 2 …… (1) Partially differentiate the function for x , f x = ∂ ∂ x ( x 4 + y 4 + 4 x − 32 y + 80 ) = ∂ ∂ x ( x 4 ) + ∂ ∂ x ( y 4 ) + ∂ ∂ x ( 4 x ) − ∂ ∂ x ( 32 y ) + ∂ ∂ x ( 80 ) = 4 x 3 − 0 + 4 − 0 + 0 = 4 x 3 + 4 …… (2) Partially differentiate the function (1) for y . f y = ∂ ∂ y ( x 4 + y 4 + 4 x − 32 y + 80 ) = ∂ ∂ y ( x 4 ) + ∂ ∂ y ( y 4 ) + ∂ ∂ y ( 4 x ) − ∂ ∂ y ( 32 y ) + ∂ ∂ y ( 80 ) = 0 + 4 y 3 + 0 − 32 + 0 = 4 y 3 − 32 …… (3) Perform partial differentiation with respect to x on the function (2). f x x = ∂ ∂ x ( 4 x 3 + 4 ) = ∂ ∂ x ( 4 x 3 ) + ∂ ∂ x ( 4 ) = 12 x 2 + 0 = 12 x 2 Perform partial differentiation with respect to y on the function (3)

Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

11th Edition

ISBN: 9780133795561

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 6, Problem 26RE

To determine

To calculate: The relative maximum and minimum values for the function f(x,y)=x4+y4+4x−32y+80.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6, Problem 25RE

Chapter 6, Problem 27RE

Students have asked these similar questions

A 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.

Explain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)

use Integration by Parts to derive 12.6.1

Chapter 6 Solutions

Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

Ch. 6.1 - Forf(x,y)=x23xy,find(0,2),f(2,3),andf(10,5).Ch. 6.1 - 2. . Ch. 6.1 - 3. . Ch. 6.1 - 4. . Ch. 6.1 - Forf(x,y)=Inx+y3,findf(e,2),f(e2,4),andf(e3,5).Ch. 6.1 - 6. . Ch. 6.1 - Forf(x,y,z)=x2y2+z2,findf(1,2,3)andf(2,1,3).Ch. 6.1 - 8. . Ch. 6.1 - In Exercises 9-14, determine the domain of each...Ch. 6.1 - In Exercises 9-14, determine the domain of each...

Additional Math Textbook Solutions

Find more solutions based on key concepts

Find the polar coordinates, 0 ≤ θ ≤ 2π and r ≥ 0, of the following points given in Cartesian coordinates. (1, 1...

University Calculus: Early Transcendentals (4th Edition)

Identifying Type I and Type II Errors In Exercises 31–36, describe type I and type II errors for a hypothesis t...

Elementary Statistics: Picturing the World (7th Edition)

Explain the meaning of the term “statistically significant difference” in statistics terminology.

Intro Stats, Books a la Carte Edition (5th Edition)

2. Source of Data In conducting a statistical study, why is it important to consider the source of the data?

Elementary Statistics

CHECK POINT I You deposit $3000 in s savings account at Yourtown Bank, which has rate of 5%. Find the interest ...

Thinking Mathematically (6th Edition)

Fill in each blank so that the resulting statement is true. The quadratic function f(x)=a(xh)2+k,a0, is in ____...

Algebra and Trigonometry (6th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The relative maximum and minimum values for the function f ( x , y ) = x 4 + y 4 + 4 x − 32 y + 80 .

Solution Summary: The author calculates the relative maximum and minimum values for the function f(x,y).