Explanation Given Information: The provided function is: f ( x ) = x + 1 ( x − 1 ) 2 Formula used: First Derivative Test, suppose that c is a critical point of the continuous function f and its derivative is defined for x close to, and on both sides of x = c . Then, determine the sign of the derivative to the left and the right of x = c . If f ′ ( x ) is positive to the left of x = c and negative to the right, then f has a maximum at x = c . If f ′ ( x ) is negative to the left of x = c and positive to the right, then f has a minimum at x = c . If f ′ ( x ) has the same sign on both sides of x = c , then f has neither a maximum nor a minimum at x = c . Stationary Points are the points in the interior of the domain where the derivative is zero. To locate stationary points set f ′ ( x ) = 0 and solve for x . Singular Points are the points in the interior of the domain where the derivative is not defined. To locate singular points, find the interior points x where f ′ ( x ) is not defined but f ( x ) is defined. Closed intervals contain end points but open intervals do not have end points. Calculation: Consider the provided function, f ( x ) = x + 1 ( x − 1 ) 2 To calculate the relative and absolute extrema first, differentiating both sides of the function with respect to x as, f ′ ( x ) = [ 1 ⋅ ( x − 1 ) 2 − 2 ( x + 1 ) ( x − 1 ) ( 1 ) ( x − 1 ) 4 ] = [ ( x − 1 ) { ( x − 1 ) − 2 ( x + 1 ) } ( x − 1 ) 4 ] = [ ( x − 1 − 2 x − 2 ) ( x − 1 ) 3 ] = − x − 3 ( x − 1 ) 3 Locate stationary points: Solve the equation f ′ ( x ) = 0 to get,

Student Solutions Manual for Waner/Costenoble's Applied Calculus, 7th

7th Edition

ISBN: 9781337291293

Author: Waner, Stefan; Costenoble, Steven

Publisher: Brooks Cole

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Chapter 5, Problem 4RE

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To calculate: The exact location of the relative and absolute extrema of the function f(x)=x+1(x−1)2 for −2≤x≤2 where, x≠1.

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Chapter 5, Problem 3RE

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The exact location of the relative and absolute extrema of the function f ( x ) = x + 1 ( x − 1 ) 2 for − 2 ≤ x ≤ 2 where, x ≠ 1 .

Solution Summary: The author calculates the exact location of relative and absolute extrema of the function f(x)=x+1 left (x-1 )2.

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To calculate: The exact location of the relative and absolute extrema of the function f(x)=x+1(x−1)2 for −2≤x≤2 where, x≠1.

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Chapter 5 Solutions