Explanation Given: The function is g ( x ) = x 2 ln x . Calculation: Domain : The domain is the set of all real numbers for which y exists. It is noted that x 2 > 0 and ln x > 0 . Hence, x 2 ln x > 0 . Therefore, the domain of the function is ( 0 , ∞ ) . Symmetry : Consider g ( x ) = x 2 ln x . Obtain the odd and even symmetry by substituting f ( x ) = f ( − x ) in the above. g ( − x ) = ( − x ) 2 ln ( − x ) = − x 2 ln ( x ) Hence, it is neither odd function nor even function. Thus, for the above function it has no symmetry. Derivatives : The derivative of the above function g ( x ) = x 2 ln x as follows, g ′ ( x ) = d d x ( x 2 ln x ) = 2 x ln x + x 2 ( 1 x ) = 2 x ln x + x Find the second derivative of the above function, g ″ ( x ) = d d x ( 2 x ln x + x ) = 2 ( ln x + x 1 x ) + 1 = 2 ln x + 2 + 1 = 2 ln x + 3 Increasing/decreasing and concavity : Put g ′ ( x ) = 0 . 2 x ln x + x = 0 x ( 2 ln x + 1 ) = 0 x = not true and 2 ln x + 1 = 0 2 ln x = − 1 Further simplification, ln x = − 1 2 e ln x = e − 1 2 x = 1 e Substitute x = 1 2 in g ′ ( x ) . g ′ ( 1 2 ) = 2 ln 1 2 + 1 2 = − 0.886 Hence, the function decreases at above intervals at ( 0 , 1 e ) . Therefore, the function is increasing on ( 1 e , ∞ ) , and the function decreasing on ( 0 , 1 e )

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 4.7, Problem 106E

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Chapter 4.7, Problem 105E

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Consider the function f(x) = x3. Where is the...Ch. 4.1 - Sketch the graph of a function that is continuous...Ch. 4.1 - Sketch the graph of a function that has an...Ch. 4.1 - What is a critical point of a function?Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Sketch the graph of a function f that has a local...Ch. 4.1 - Absolute maximum/minimum values Use the following...Ch. 4.1 - Prob. 12E Ch. 4.1 - Absolute maximum/minimum values Use the following...

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The graph of the function g ( x ) = x 2 ln x .

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