Explanation Given: The given function is: f ( x , y ) = 4 x 3 + 3 x y + 4 y 3 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: f ( x , y ) = 4 x 3 + 3 x y + 4 y 3 First find the critical points of each function: f x = ∂ ∂ x ( 4 x 3 + 3 x y + 4 y 3 ) = 12 x 2 + 3 y and, f y = ∂ ∂ y ( 4 x 3 + 3 x y + 4 y 3 ) = 3 x + 12 y 2 Equate to zero, we get: 12 x 2 + 3 y = 0 3 y = − 12 x 2 y = − 4 x 2 and, 3 x + 12 ( − 4 x 2 ) 2 = 0 x ( 1 + 4 3 x 3 ) = 0 x = 0 , − 1 4 The corresponding value of y is: y = − 4 ( 0 ) 2 = 0 and, y = − 4 ( − 1 4 ) 2 = − 1 4 Therefore, the critical points are ( 0 , 0 ) and ( − 1 4 , − 1 4 )

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

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 9.3, Problem 2YT

To determine

To find:

The point at which the function has relative maximum, relative minimum or saddle point 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 1YT

Chapter 9.3, Problem 1E

Students have asked these similar questions

纟 W₂ Find S-FX SB.MX

4. The revenue (in thousands of dollars) from producing x units of an item is R(x)=8x-0.015 x². a) Find the average rate of change of revenue when the production is increased from 1000 to 1001 units.

MATH 122 WORKSHEET 3 February 5, 2025 . Solve the following problems on a separate sheet. Justify your answers to earn full credit. 1. Let f(x) = x² - 2x + 1. (a) Find the slope of the graph of y = f (x) at the point P = (0,1) by directly evaluating the limit: f'(0) = lim ( f(Ax) - f(0) Ax Ax→0 (b) Find the equation of the tangent line 1 to the graph of ƒ at P. What are the x and y intercepts of 1 ? (c) Find the equation of the line, n, through P that is perpendicular to the tangent line l. (Line n is called the normal line to the graph of f at P.) (d) Sketch a careful graph that displays: the graph of y = f (x), its vertex point, its tangent and normal lines at point P, and the x and y intercepts of these lines. Bonus: Find the coordinates of the second point, Q, (QP), at which the normal line n intersects the graph of f. 2. A rock is thrown vertically upward with an initial velocity of 20 m/s from the edge of a bridge that is 25 meters above a river bed. Based on Newton's Laws of…

Chapter 9 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

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 at which the function has relative maximum , relative minimum or saddle point for the given function.

Solution Summary: The author explains how to find the point at which the function has relative maximum, relative minimum, or saddle point for the given function.

Concept explainers

Videos

To find:

The point at which the function has relative maximum, relative minimum or saddle point for the given function.

Want to see the full answer?

Chapter 9 Solutions

The point at which the function has relative maximum , relative minimum or saddle point for the given function.

Solution Summary: The author explains how to find the point at which the function has relative maximum, relative minimum, or saddle point for the given function.

Concept explainers

Videos

To find: The point at which the function has relative maximum, relative minimum or saddle point for the given function.

Want to see the full answer?

Chapter 9 Solutions

To find:

The point at which the function has relative maximum, relative minimum or saddle point for the given function.