Explanation Given information: The function f ( x ) is, f ( x ) = x 5 − 6 x 4 + 14 x 3 − 16 x 2 + 9 x + 12 Here, the value of c = 1 . Concept used: Higher Derivative Test: Consider all the derivative of f at x = c to be zero except n th derivative of f . If n is even, then f ( c ) is a local minimum when f ( n ) ( c ) > 0 and a local maximum when f ( n ) ( c ) < 0 . Calculation: The derivative of f can be calculated as, d d x f ( x ) = d d x ( x 5 − 6 x 4 + 14 x 3 − 16 x 2 + 9 x + 12 ) f ′ ( x ) = 5 x 4 − 24 x 3 + 42 x 2 − 32 x + 9 It is given that c = 1 , the obtain f ′ ( c ) as, f ′ ( c ) = f ′ ( 1 ) = 5 ( 1 ) 4 − 24 ( 1 ) 3 + 42 ( 1 ) 2 − 32 ( 1 ) + 9 = 5 − 24 + 42 − 32 + 9 = 0 Therefore, x = 1 is a critical point. The second derivative of the function f ( x ) can be calculated as, d d x f ′ ( x ) = d d x ( 5 x 4 − 24 x 3 + 42 x 2 − 32 x + 9 ) f ″ ( x ) = 20 x 3 − 72 x 2 + 84 x − 32 Substitute 1 for x in second derivative of f ( x ) as, f ″ ( 1 ) = 20 ( 1 ) 3 − 72 ( 1 ) 2 + 84 ( 1 ) − 32 = 20 − 72 + 84 − 32 = 0 The second derivative is equal to zero, so obtain the next derivative and check at x = 1 To determine (b) To check: Whether x = 0 is a critical point of the function f ( x ) = x 6 − x 3 or not and f ( 0 ) is a local minimum or a local maximum.

4th Edition

ISBN: 9781319055905

Author: FRANZOSA

Publisher: VST

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Question

Chapter 4.5, Problem 84E

To determine

(a)

To check:

Whether x=1 is a critical point of the function f(x)=x5−6x4+14x3−16x2+9x+12 or not and f(1) is a local minimum or a local maximum.

To determine

(b)

To check:

Whether x=0 is a critical point of the function f(x)=x6−x3 or not and f(0) is a local minimum or a local maximum.

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Chapter 4.5, Problem 83E

Chapter 4.6, Problem 1PQ

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Chapter 4 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

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(a) To check: Whether x = 1 is a critical point of the function f ( x ) = x 5 − 6 x 4 + 14 x 3 − 16 x 2 + 9 x + 12 or not and f ( 1 ) is a local minimum or a local maximum .

Solution Summary: The author evaluates whether x=1 is a critical point of the function f(x) or not.

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(a)

To check:

Whether x=1 is a critical point of the function f(x)=x5−6x4+14x3−16x2+9x+12 or not and f(1) is a local minimum or a local maximum.

(b)

To check:

Whether x=0 is a critical point of the function f(x)=x6−x3 or not and f(0) is a local minimum or a local maximum.

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Chapter 4 Solutions

(a) To check: Whether x = 1 is a critical point of the function f ( x ) = x 5 − 6 x 4 + 14 x 3 − 16 x 2 + 9 x + 12 or not and f ( 1 ) is a local minimum or a local maximum .

Solution Summary: The author evaluates whether x=1 is a critical point of the function f(x) or not.

Concept explainers

Videos

(a) To check: Whether x=1 is a critical point of the function f(x)=x5−6x4+14x3−16x2+9x+12 or not and f(1) is a local minimum or a local maximum.

(b) To check: Whether x=0 is a critical point of the function f(x)=x6−x3 or not and f(0) is a local minimum or a local maximum.

Want to see the full answer?

Chapter 4 Solutions

(a)

To check:

Whether x=1 is a critical point of the function f(x)=x5−6x4+14x3−16x2+9x+12 or not and f(1) is a local minimum or a local maximum.

(b)

To check:

Whether x=0 is a critical point of the function f(x)=x6−x3 or not and f(0) is a local minimum or a local maximum.