Explanation Given: The given function is, f ( x ) = 2 x − 5 x + 3 . Approach: The critical points ( x , f ( x ) ) can be find by substituting 0 for f ′ ( x ) and finding value of x and substituting those values in the function. The inflection points of a function are the points ( x , f ( x ) ) for which f ″ ( x ) = 0 . The x intercept of the function are the points ( x , f ( x ) ) for which f ( x ) = 0 . The y intercept of the function are the points ( 0 , f ( 0 ) ) for which x = 0 . Calculation: Consider the given function, f ( x ) = 2 x − 5 x + 3 ... ( 1 ) . The above function exists for all values of x for which denominator is non-zero. So the function does not exists if, x + 3 = 0 x = − 3 So domain of function is ( − ∞ , − 3 ) ∪ ( − 3 , ∞ ) . The limit of the function is, lim x → − 3 f ( x ) = ∓ ∞ So, the vertical line x = − 3 is an asymptote of the function. And, lim x → ∓ ∞ f ( x ) = 2 So, the horizontal line y = 2 is an asymptote of the function. Differentiate the above function with respect to x . f ′ ( x ) = ( x + 3 ) ( 2 ) − ( 2 x − 5 ) ( 1 ) ( x + 3 ) 2 = 2 x + 6 − 2 x + 10 ( x + 3 ) 2 = 16 ( x + 3 ) 2 Substitute 0 for f ( x ) in above equation to find critical points. 16 ( x + 3 ) 2 = 0 The above equation can never be zero, so the function has no critical points

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 5.CR, Problem 50CR

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The function considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

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To graph: The function considering domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points, regions where function is concave up or concave down, intercepts where possible, and asymptotes where applicable.

Solution Summary: The author explains how the function considers domain, critical points, symmetry, regions where function is increasing or decreasing, inflection points and asymptotes where applicable.