Explanation Given information: The provided function is f ( x ) = x 4 − 4 x 3 . Formula used: Stationary Points: Points in the interior of the domain where the derivative is zero and to locate stationary points set f ′ ( x ) = 0 and solve for x . Second derivative test: If the function f has a stationary point at c and that f ″ ( c ) exists then determine the sign of f ″ ( c ) : 1 : If f ″ ( c ) > 0 , then f has a relative minimum at c . 2: If f ″ ( c ) < 0 , then f has a relative maximum at c . 3: If f ″ ( c ) = 0 then the test is inconclusive and uses the first derivative test. 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 . Calculation: The provided function is: f ( x ) = x 4 − 4 x 3 Differentiate the above equation with respect to x : f ′ ( x ) = d d x ( x 4 − 4 x 3 ) = 4 x 3 − 12 x 2 Consider the formula for stationary points: f ′ ( x ) = 0 Substitute 4 x 3 − 12 x 2 f ' ( x ) to get: 4 x 3 − 12 x 2 = 0 Solve the above equation x to get: 4 x 3 − 12 x 2 = 0 x 2 ( 4 x − 12 ) = 0 Thus, x 2 = 0 x = 0 And 4 x − 12 = 0 4 x = 12 x = 3 Since, f ′ ( x ) = 0 at x = 0 , 3

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7th Edition

ISBN: 9781337652742

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.3, Problem 39E

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To calculate: The x− coordinates of all the critical points of the function f(x)=x4−4x3 and check whether each critical point is relative maximum, relative minimum or neither by first applying the second derivative test and if the test fails, by some other method.

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Chapter 5.3, Problem 38E

Chapter 5.3, Problem 40E

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The x − coordinates of all the critical points of the function f ( x ) = x 4 − 4 x 3 and check whether each critical point is relative maximum , relative minimum or neither by first applying the second derivative test and if the test fails, by some other method.

Solution Summary: The author calculates the x- coordinates of all the critical points of the function and checks whether each critical point is relative maximum, relative minimum or neither.

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To calculate: The x− coordinates of all the critical points of the function f(x)=x4−4x3 and check whether each critical point is relative maximum, relative minimum or neither by first applying the second derivative test and if the test fails, by some other method.

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