Explanation If the derivative of function f ′ ( x ) changes sign from positive to negative as the function’s graph move across a critical number c , then function f has relative maximum at x = c . If the derivative of function f ′ ( x ) changes sign from negative to positive as the function’s graph move across a critical number c , then function f has relative minimum at x = c . The given function is, f ( x ) = 4 x 3 − 4 x 2 + x + 10 . Plot the graph of given function in standard viewing window. To graph an equation, follow the steps using graphing calculator. First press “ON” button on graphical calculator, press Y = key and enter right hand side of the equation f ( x ) = 4 x 3 − 4 x 2 + x + 10 after the symbol Y 1 . Enter the keystrokes 4 X ^ ( 3 ) − 4 X x 2 + X + 10 . The display will show the equation, Y 1 = 4 X^ ( 3 ) − 4 X 2 + X + 10 . The settings for standard viewing window are [ − 10 , 10 ] by [ − 10 , 10 ] . Now, press the GRAPH key to produce the graph in standard window as shown in Figure 1. Figure 1 Since, the domain of function f is the interval ( − ∞ , ∞ ) . Here, it can be observed that Figure 1 does not show the right side of the graph of function. So, enlarge the viewing window accordingly. To get the right side of the graph of function and plot the graph of given function in viewing window [ − 10 , 10 ] × [ − 10 , 20 ] . Press the WINDOW key to change the window settings. X min = − 10 X max = 10 Y min = − 10 Y max = 20 Now, press the GRAPH key to produce the graph as shown in Figure 2. Figure 2 To analyze the function first find the derivative of function f ( x ) = 4 x 3 − 4 x 2 + x + 10 . d d x f ( x ) = d d x ( 4 x 3 − 4 x 2 + x + 10 ) f ′ ( x ) = 12 x 2 − 8 x + 1 = 12 x 2 − 6 x − 2 x + 1 = 6 x ( 2 x − 1 ) − 1 ( 2 x − 1 ) Further solve the above equation to get the critical points

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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To plot: The graphs of given function and then analyze it.

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