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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.5, Problem 7E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Applying the Second-Partials Test In Exercises 1-18, find the relative extrema and saddle points of the function. See Examples

f ( x , y ) = ( 3 x 2 + 6 y 2 + 8 ) 1 / 3

To determine

To calculate: The relative extrema and saddle point of the function f(x,y)=(3x2+6y2+8)13.

Explanation

Given Information:

The given function is f(x,y)=(3x2+6y2+8)13.

Formula used:

Partial derivatives of the function with respect to x as well as y,

fx=∂f(x,y)∂xfy=∂f(x,y)∂y

Double partial derivatives of the function,

fxx=∂2f(x,y)∂x2fyy=∂2f(x,y)∂y2fxy=∂2f(x,y)∂x∂y

At the critical point if the value of D=fxx(a,b)fyy(a,b)−[fxy(a,b)]2,

D>0 and r<0, then declare the point as a maxima.

D>0 and r>0, then declare the point as a minima.

D<0, then this critical point will not be a extremum point.

D=0, then say double partial derivative fails and need further investigation on this point.

Calculation:

Consider the given equation

f(x,y)=(3x2+6y2+8)13

The critical point of the function f(x,y) is given by

∂f∂x=0   and   ∂f∂y=0

Now, find the partial derivative with respect to x,

∂f∂x=013(3x2+6y2+8)−23(6x)=0x=0

Now, find the partial derivative with respect to y,

∂f∂y=013(3x2+6y2+8)−23(12y)=0y=0

Hence, the critical point is (0,0).

Now, the second derivatives of f(x,y) at (0,0) with respect to x,

fxx=∂∂x(2x(3x2+6y2+8)23)=2(3x2+6y2+8)23−23(2×6x2(3x2+6y2+8)13)(3x2+6y2+8)43=2(3x2+6y2+8)−8x2(3x2+6y2+8</

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

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For Problems 5-54, perform the following operations with real numbers. Objectives 3-6 (19)0

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In Exercises 23 to 40, let U {p, q, r, s, t}, D = (p. r, s, t}, E = {q, s), F = {p, t}, and G = {s}. Determine ...

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Mathematical Applications for the Management, Life, and Social Sciences

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Probability and Statistics for Engineering and the Sciences

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Contemporary Mathematics for Business & Consumers

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Finite Mathematics for the Managerial, Life, and Social Sciences

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Mathematics: A Practical Odyssey

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Find the vertical asymptotes, if any, of each function. a. g(x)=x+5x225b. h(x)=5x2+25

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Solve the following exercises. For customary unit-dimensioned exercises, calculate angles to the nearer minute ...

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