Explanation Given information: f ( x , y ) = x y e − x 2 − y 2 Formula used: Second derivative test: If P = ( a , b ) is a critical point of f ( x , y ) , then D ( a , b ) > 0 , f x x ( a , b ) > 0 ⇒ f ( a , b ) is a local minimum point D ( a , b ) > 0 , f x x ( a , b ) < 0 ⇒ f ( a , b ) is a local maximum point D ( a , b ) < 0 ⇒ f ( a , b ) is a saddle point D ( a , b ) < 0 ⇒ test fails Where, D = f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 Calculation: To find the critical points of f ( x , y ) = x y e − x 2 − y 2 the formula ∇ f = ( f x , f y ) = ( 0 , 0 ) Thus, ∇ f = ( d ( x y e − x 2 − y 2 ) d x , d ( x y e − x 2 − y 2 ) d y ) = ( 0 , 0 ) ∇ f = ( e − x 2 − y 2 ( 1 − 2 x 2 ) , e − x 2 − y 2 ( 1 − 2 y 2 ) ) = ( 0 , 0 ) ⇒ e − x 2 − y 2 ( 1 − 2 x 2 ) = 0 & e − x 2 − y 2 ( 1 − 2 y 2 ) = 0 ⇒ ( 1 − 2 x 2 ) = 0 & ( 1 − 2 y 2 ) = 0 ⇒ x = ± 1 2 & y = ± 1 2 Thus critical points are ( 1 2 , 1 2 ) , ( − 1 2 , 1 2 ) , ( 1 2 , − 1 2 ) , ( − 1 2 , − 1 2 ) Now to find if they are local minimum, local maximum or saddle points by the use of second derivative test First to find D, find f x x , f y y & f x y D = f x x ( a , b ) f y y ( a , b ) − ( f x y ( a , b ) ) 2 Thus, f x x = d d x f x = d d x ( ( 1 − 2 x 2 ) e − x 2 − y 2 ) = e − x 2 − y 2 ( 4 x 3 − 6 x ) f y y = d d y f y = d d y ( ( 1 − 2 y 2 ) e − x 2 − y 2 ) = e − x 2 − y 2 ( 4 y 3 − 6 y ) , and f x y = d d x f y = d d x ( ( 1 − 2 y 2 ) e − x 2 − y 2 ) = e − x 2 − y 2 ( − 2 x + 2 x y 2 ) Now, check for ( a , b ) = ( 1 2 , 1 2 ) , f x x ( 1 2 , 1 2 ) = e − ( 1 2 ) 2 − ( 1 2 ) 2 ( 4 ( 1 2 ) 3 − 6 ( 1 2 ) ) = e − 1 ( − 4 2 ) ≺ 0 f y y ( 1 2 , 1 2 ) = e − ( 1 2 ) 2 − ( 1 2 ) 2 ( 4 ( 1 2 ) 3 − 6 ( 1 2 ) ) = e − 1 ( − 4 2 ) ≺ 0 f x y ( 1 2 , 1 2 ) = e − ( 1 2 ) 2 − ( 1 2 ) 2 ( − 2 ( 1 2 ) + 2 ( 1 2 ) ( 1 2 ) 2 ) = e − 1 ( − 1 2 ) ≺ e − 1 ( − 4 2 ) ≺ 0 Thus, D = f x x ( 1 2 , 1 2 ) f y y ( 1 2 , 1 2 ) − ( f x y ( 1 2 , 1 2 ) ) 2 D = ( e − 1 ( − 4 2 ) ) 2 − ( e − 1 ( − 1 2 ) ) 2 ≻ 0 As, D > 0 , f x x ( 1 2 , 1 2 ) < 0 thus, function has local maxima at ( 1 2 , 1 2 ) Now for ( a , b ) = ( − 1 2 , 1 2 ) , f x x ( − 1 2 , 1 2 )

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

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 14.7, Problem 15E

To determine

Find critical points of f(x,y)=xye−x2−y2, then find whether they are local minimum, local maximum or saddle points.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.7, Problem 14E

Chapter 14.7, Problem 16E

Students have asked these similar questions

Can i get help with #26 ?

3. Let f(x) = 8x- 3x*+5 %3! a) Find where f has stationary points. b) Perform a sign analysis on the derivative of f. Find where f is increasing and decreasing. c) Use your results to find where all local maxima and minima of foccur.

Find all critical numbers of the function y= minimum. x²-9 X-5 x#5. Then use the second-derivative test on each critical number to determine whether it leads to a local maximum or

Chapter 14 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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.

Find critical points of f ( x , y ) = x y e − x 2 − y 2 , then find whether they are local minimum , local maximum or saddle points.

Concept explainers

Videos

Find critical points of f(x,y)=xye−x2−y2, then find whether they are local minimum, local maximum or saddle points.

Want to see the full answer?

Chapter 14 Solutions