Explanation Given: The given function is: z = − 2 x 3 − 3 y 4 + 6 x y 2 + 1 16 Approach: For a function z = f ( x , y ) , let f x x , f y y , and f x y all exist in a circular region contained in the xy -plane with center ( a , b ) . Further, let f x ( a . b ) = 0 and f y ( a . b ) = 0 . Define the number D , known as the discriminant, by D = f x x ( a , b ) ⋅ f y y ( a , b ) − [ f x y ( a , b ) ] 2 Then, a. f ( a , b ) is a relative maximum if D > 0 and f x x ( a , b ) < 0 . b. f ( a , b ) is a relative minimum if D > 0 and f x x ( a , b ) > 0 . c. f ( a , b ) is a saddle point (neither a maximum nor a minimum) if D < 0 . d. if D = 0 , the test gives no information. Calculation: Consider the given function, z = − 2 x 3 − 3 y 4 + 6 x y 2 + 1 16 Find the f x ( x , y ) , and f y ( x , y ) : f x ( x , y ) = ∂ ∂ x ( − 2 x 3 − 3 y 4 + 6 x y 2 + 1 16 ) = − 6 x 2 + 6 y 2 and, f y ( x , y ) = ∂ ∂ y ( − 2 x 3 − 3 y 4 + 6 x y 2 + 1 16 ) = − 12 y 3 + 12 x y A function has relative extrema at a point, if f x ( x , y ) = 0 − 6 x 2 + 6 y 2 = 0 And if f y ( x , y ) = 0 , f y ( x , y ) = 0 − 12 y 3 + 12 x y = 0 Solve the equation − 6 x 2 + 6 y 2 = 0 and − 12 y 3 + 12 x y = 0 , we get: x = 0 , y = 0 , x = 1 , y = ± 1 . Therefore, ( 0 , 0 ) , ( 1 , − 1 ) and ( 1 , 1 ) are the critical point where extrema can occur. Check for saddle point: f x y ( x , y ) = ∂ ∂ x ( − 12 y 3 + 12 x y ) = 12 y , f x x ( x , y ) = ∂ ∂ x ( − 6 x 2 + 6 y 2 ) = − 12 x , and,

2nd Edition

ISBN: 9780321964458

Author: Lial

Publisher: PEARSON EDUCATION (COLLEGE)

See similar textbooks

Concept explainers

Transformation of Graphs

The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.

Exponential Functions

The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, c…

Videos

Question

Chapter 9.3, Problem 24E

To determine

To find:

The point for relative extrema and then match the equation to its graph for the given function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 23E

Chapter 9.3, Problem 25E

Chapter 9 Solutions

EBK CALCULUS FOR THE LIFE SCIENCES

Ch. 9.1 - YOUR TURN For the function in Example 1, find...Ch. 9.1 - Prob. 2YT Ch. 9.1 - Prob. 3YT Ch. 9.1 - EXERCISES Let f(x,y)=2x3y+5.Find the following. a....Ch. 9.1 - Prob. 2E Ch. 9.1 - EXERCISES Let h(x,y)=x2+2y2. Find the following....Ch. 9.1 - EXERCISES Let f(x,y)=9x+5ylogx. Find the...Ch. 9.1 - EXERCISES Let f(x,y)=ex+ln(x+y). Find the...Ch. 9.1 - EXERCISES Let f(x,y)=xex+y. Find the following. a....Ch. 9.1 - EXERCISES Let f(x,y)=xsin(x2y). Find the...

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 point for relative extrema and then match the equation to its graph for the given function.

Solution Summary: The author explains how to find the point for relative extrema and match the equation to its graph for the given function.

Concept explainers

Videos

To find: The point for relative extrema and then match the equation to its graph for the given function.

Want to see the full answer?

Chapter 9 Solutions

To find:

The point for relative extrema and then match the equation to its graph for the given function.