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

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 12.3, Problem 39E

To determine

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 12 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

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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.

Chapter 12 Solutions