Explanation Proof: The given function is f ( x ) = x 7 / 3 . The first derivative of the function f is, f ′ ( x ) = d d x ( x 7 / 3 ) = 7 3 x 7 / 3 − 1 ( d d x ( x ) n = n x n − 1 ) = 7 3 x 4 / 3 The first derivative of function f exists for all values of x as there is no value of x that can make the f ′ ( x ) undefined. Substitute 0 for x in f ′ ( x ) = 7 3 x 4 / 3 . f ′ ( 0 ) = 7 3 ( 0 ) 4 / 3 = 0 The given function has first order derivatives at all points including x = 0 . The second derivative of the function f is, f ″ ( x ) = d d x f ′ ( x ) = d d x 7 3 x 4 / 3 = 7 3 d d x x 4 / 3 ( d d x [ c f ( x ) ] = c d d x [ f ( x ) ] ) = 7 3 [ 4 3 x 4 / 3 − 1 ] ( d d x ( x ) n = n x n − 1 ) Simplify it further. f ″ ( x ) = 7 3 [ 4 3 x 1 / 3 ] = 28 9 x 1 / 3 The second derivative of function f exists for all values of x as there is no value of x that can make the f ″ ( x ) undefined

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 3.5, Problem 48E

To determine

To show: The given function has first and second order derivatives at all points including x=0 and the third derivative of the function does not exist at x=0 .

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Chapter 3.5, Problem 47E

Chapter 3.5, Problem 49E

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Chapter 3 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

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The given function has first and second order derivatives at all points including x = 0 and the third derivative of the function does not exist at x = 0 .

Solution Summary: The author explains that the given function has first and second order derivatives at all points including x=0.

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To show: The given function has first and second order derivatives at all points including x=0 and the third derivative of the function does not exist at x=0 .

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Chapter 3 Solutions