Explanation Given information: The function is, f ( x , y ) = x 2 + x y + y 2 Calculation: Consider the function, f ( x , y ) = x 2 + x y + y 2 ............ (1) Partial derivative of equation (1) with respect to x . f x = d d x ( x 2 + x y + y 2 ) = 2 x + y Partial derivative of equation (1) with respect to y . f y = d d x ( x 2 + x y + y 2 ) = x + 2 y Both f x and f y are equal at ( 0 , 0 ) so, the critical point is ( 0 , 0 ) . Consider the curve, x 2 + y 2 = 1 y = 1 − x 2 Substitute 1 − x 2 for y in equation (1). f ( x , y ) = x 2 + x ( 1 − x 2 ) + ( 1 − x 2 ) 2 = x 2 + x ( 1 − x 2 ) + ( 1 − x 2 ) = x ( 1 − x 2 ) + 1 ............ (2) Differentiate equation (2) with respect to x . f ′ ( x ) = x − 2 x 2 1 − x 2 + 1 − x 2 = − x 2 + 1 − x 2 1 − x 2 = − 2 x 2 + 1 1 − x 2 ............ (3) Substitute 0 for f ′ ( x ) in equation (3). 0 = − 2 x 2 + 1 1 − x 2 − 2 x 2 + 1 = 0 x = ± 1 2 So, y = ± 1 2 So, the critical points are ( 1 2 , 1 2 ) , ( 1 2 , − 1 2 ) , ( − 1 2 , 1 2 ) ( − 1 2 , − 1 2 )

Student's Solution and Survival Manual for Calculus

7th Edition

ISBN: 9781524934040

Author: STRAUSS MONTY J, TODA MAGDALENA DANIELE, SMITH KARL J

Publisher: Kendall Hunt Publishing

See similar textbooks

Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Videos

Question

Chapter 11.7, Problem 23PS

To determine

The absolute extrema of f on the closed bounded set.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.7, Problem 22PS

Chapter 11.7, Problem 24PS

Students have asked these similar questions

f'(x)

A body of mass m at the top of a 100 m high tower is thrown vertically upward with an initial velocity of 10 m/s. Assume that the air resistance FD acting on the body is proportional to the velocity V, so that FD=kV. Taking g = 9.75 m/s2 and k/m = 5 s, determine: a) what height the body will reach at the top of the tower, b) how long it will take the body to touch the ground, and c) the velocity of the body when it touches the ground.

A chemical reaction involving the interaction of two substances A and B to form a new compound X is called a second order reaction. In such cases it is observed that the rate of reaction (or the rate at which the new compound is formed) is proportional to the product of the remaining amounts of the two original substances. If a molecule of A and a molecule of B combine to form a molecule of X (i.e., the reaction equation is A + B ⮕ X), then the differential equation describing this specific reaction can be expressed as: dx/dt = k(a-x)(b-x) where k is a positive constant, a and b are the initial concentrations of the reactants A and B, respectively, and x(t) is the concentration of the new compound at any time t. Assuming that no amount of compound X is present at the start, obtain a relationship for x(t). What happens when t ⮕∞?

Chapter 11 Solutions

Student's Solution and Survival Manual for Calculus

Ch. 11.1 - Prob. 1PS Ch. 11.1 - Prob. 2PS Ch. 11.1 - Prob. 3PS Ch. 11.1 - Prob. 4PS Ch. 11.1 - Prob. 5PS Ch. 11.1 - Prob. 6PS Ch. 11.1 - Prob. 7PS Ch. 11.1 - Prob. 8PS Ch. 11.1 - Prob. 9PS Ch. 11.1 - Prob. 10PS

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 absolute extrema of f on the closed bounded set.

Solution Summary: The author explains the absolute extrema of f on the closed bounded set.

Concept explainers

Videos

The absolute extrema of f on the closed bounded set.

Want to see the full answer?

Chapter 11 Solutions