Explanation Theorem used: If ( c , f ( c ) ) is the inflection point of f , then c is the partition number for f . Calculation: The inflection point is a point on the graph of the function where the concavity changes. That is, the point where f ′ ′ ( x ) change its sign from positive to negative or vice versa. Since the given function is in the form of ln ( x ) , it is defined only for all x > 0 . That is, x 2 − 2 x + 5 > 0 Then, the domain of the function is ( − ∞ , ∞ ) . Obtain the second derivative of the given function in order to get the inflection point. f ′ ′ ( x ) = d 2 d x 2 ( ln ( x 2 − 2 x + 5 ) ) = d d x ( d d x ( ln ( x 2 − 2 x + 5 ) ) ) = d d x ( 1 x 2 − 2 x + 5 ( 2 x − 2 ) ) = d d x ( 2 x − 2 x 2 − 2 x + 5 ) On further simplification, f ′ ′ ( x ) = ( x 2 − 2 x + 5 ) d d x ( 2 x − 2 ) − ( 2 x − 2 ) d d x ( x 2 − 2 x + 5 ) ( x 2 − 2 x + 5 ) 2 = ( x 2 − 2 x + 5 ) ( 2 ) − ( 2 x − 2 ) ( 2 x − 2 ) ( x 2 − 2 x + 5 ) 2 = 2 x 2 − 4 x + 10 − ( 2 x − 2 ) 2 ( x 2 − 2 x + 5 ) 2 = 2 x 2 − 4 x + 10 − ( 4 x 2 − 8 x + 4 ) ( x 2 − 2 x + 5 ) 2 On further simplification, f ′ ′ ( x ) = 2 x 2 − 4 x + 10 − 4 x 2 + 8 x − 4 ( x 2 − 2 x + 5 ) 2 = − 2 x 2 + 4 x + 6 ( x 2 − 2 x + 5 ) 2 Equate f ′ ′ ( x ) to 0 to obtain the partition number. − 2 x 2 + 4 x + 6 ( x 2 − 2 x + 5 ) 2 = 0 − 2 x 2 + 4 x + 6 = 0 − 2 ( x 2 − 2 x − 3 ) = 0 x 2 − 2 x − 3 = 0 On further simplification, ( x − 3 ) ( x + 1 ) = 0 ( x − 3 ) and ( x + 1 ) = 0 x = 3 and x = − 1 So, the partition number is x = 3 and x = − 1 . The sign chart for f ′ ′ ( x ) = − 2 x 2 + 4 x + 6 ( x 2 − 2 x + 5 ) 2 with the partition number x = 3 and x = − 1 is as follows

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Chapter 11.2, Problem 3MP

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To examine: The inflection points of the function f(x)=ln(x2−2x+5).

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Chapter 11.2, Problem 2MP

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Pearson eText for College Mathematics for Business, Economics, Life Sciences, and Social Sciences -- Instant Access (Pearson+)

Ch. 11.1 - Matched Problem 1 Repeat Example 1 for f(x) = x2 –...Ch. 11.1 - Matches Problem 2 Find the critical numbers of f,...Ch. 11.1 - Matched Problem 3 Find the critical numbers of f,...Ch. 11.1 - Prob. 4MP Ch. 11.1 - Prob. 5MP Ch. 11.1 - Prob. 6MP Ch. 11.1 - Prob. 7MP Ch. 11.1 - Prob. 8MP Ch. 11.1 - Prob. 9MP Ch. 11.1 - Skills Warm-up Exercises In Problems 1-8, inspect...

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To examine: The inflection points of the function f ( x ) = ln ( x 2 − 2 x + 5 ) .

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