Explanation Given: The function is: f ( x ) = 2 3 x 3 − 1 2 x 2 − 3 x − 4 . Formula used: Write the formula for derivative of power rule. If f ( x ) = x n for any number n , then the derivative. f ′ ( x ) = n x n − 1 If f ( x ) = k for any real number k , then the derivative is defined by the formula. f ′ ( x ) = 0 To find local maximum and minimum find the values of x from the equation f ′ ( x ) = 0 . Then find f ″ ( x ) and put the values of x. If f ″ ( x ) equals some positive quantity then it will be Local minimum is located at that point x and if f ″ ( x ) equals some negative quantity then Local maximum is located at the point at that point. Calculation: Consider the provided function: f ( x ) = 2 3 x 3 − 1 2 x 2 − 3 x − 4 …… (1) Differentiate the function with respect to x by using the power rule of the derivative. f ′ ( x ) = d d x ( 2 3 x 3 − 1 2 x 2 − 3 x − 4 ) = 2 3 ( 3 ) x 3 − 1 − 1 2 ( 2 ) x 2 − 1 − 3 ( 1 ) x 1 − 1 − 0 = 2 x 2 − x − 3 …… (2) Calculate the critical point by equating f ′ ( x ) equal to zero. Substitute 0 for f ′ ( x ) . 2 x 2 − x − 3 = 0 Solve for x . 2 x 2 + 2 x − 3 x − 3 = 0 2 x ( x + 1 ) − 3 ( x + 1 ) = 0 ( 2 x − 3 ) ( x + 1 ) = 0 Further simplify the equation. x = − 1 and x = 3 2 The critical points are at x = − 1 and x = 3 2 . Hence, there may be three test points: At x = − 2 , x = 0 and x = 2 . Substitute − 2 for x in equation (2). f ′ ( − 2 ) = 2 ( − 2 ) 2 − ( − 2 ) − 3 = 7 Substitute 0 for x in equation (2)

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

12th Edition

ISBN: 9780134767628

Author: Margaret L. Lial, Thomas W. Hungerford, John P. Holcomb, Bernadette Mullins

Publisher: PEARSON

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Chapter 12.1, Problem 24E

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To calculate: The location of local extremum of the function f(x)=23x3−12x2−3x−4.

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Chapter 12.1, Problem 23E

Chapter 12.1, Problem 25E

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Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.

************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.

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Chapter 12 Solutions

Mathematics with Applications In the Management, Natural, and Social Sciences (12th Edition)

Ch. 12.1 - Checkpoint 1 For what values of x is the function...Ch. 12.1 - Checkpoint 2 Find all intervals on which is...Ch. 12.1 - Checkpoint 3 Identity the x-values of all points...Ch. 12.1 - Checkpoint 4 Find the critical numbers for each of...Ch. 12.1 - Prob. 5CP Ch. 12.1 - Prob. 6CP Ch. 12.1 - Checkpoint 7 Find the locations of the local...Ch. 12.1 - Prob. 8CP Ch. 12.1 - Checkpoint 9 If a sales function is given by...Ch. 12.1 - Prob. 1E

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The location of local extremum of the function f ( x ) = 2 3 x 3 − 1 2 x 2 − 3 x − 4 .

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To calculate: The location of local extremum of the function f(x)=23x3−12x2−3x−4.

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