Explanation Given: The function is, h ( x ) = 3 2 x 4 − 2 x 3 − 6 x 2 + 8 . Calculation: The function is a polynomial function, so the domain of the function is ( − ∞ , ∞ ) . Substitute 0 for x in the given function to evaluate the y -intercept. h ( 0 ) = 3 2 ⋅ 0 4 − 2 ⋅ 0 3 − 6 ⋅ 0 2 + 8 = 8 So, the y intercept is 8. Since, the function is polynomial x intercept is not defined and there is no vertical or horizontal asymptotes. Differentiate the function h and equate to 0 and evaluate the critical point. h ′ ( x ) = 0 d d x ( 3 2 x 4 − 2 x 3 − 6 x 2 + 8 ) = 0 3 2 ⋅ 4 x 3 − 2 ⋅ 3 x 2 − 6 ⋅ ( 2 x ) = 0 [ ∵ d d x x n = n x n − 1 ] 6 x 3 − 6 x 2 − 12 x = 0 Further solve the above equation. 6 x ( x 2 − x − 2 ) = 0 6 x ( x − 2 ) ( x + 1 ) = 0 ( Factors ) Equate each factor to 0 and obtain values of x . 6 x = 0 x = 0 Equate x − 2 to 0. x − 2 = 0 x = 2 Equate x + 1 to 0. x + 1 = 0 x = − 1 The critical points are, x = 0 , x = − 1 and x = 2 , so the intervals are ( − ∞ , − 1 ) , ( − 1 , 0 ) , ( 0 , 2 ) and ( 2 , ∞ ) . To check the sign of h ′ , check the test point of the function h ′ . The table below shows the test value. Intervals Test point c h ′ ( c ) Sign of h ′ ( − ∞ , − 1 ) − 2 − 48 Negative ( − 1 , 0 ) − 1 2 15 4 Positive ( 0 , 2 ) 1 − 6 Negative ( 2 , ∞ ) 3 72 Positive Sign diagram of h ′ is shown below. Figure (1) So, the sign diagram of f ′ shows that f is increasing on interval ( − 1 , 0 ) ∪ ( 2 , ∞ ) and decreasing on interval ( − ∞ , − 1 ) ∪ ( 0 , 2 ) . Figure (1) shows the function changes its sign from negative to positive at x = − 1 and x = 2 , so the relative minima occurs at x = − 1 or x = 2 and the sign changes from positive to negative at x = 0 , so the relative maxima occurs at x = 0 . Substitute − 1 for x in h ( x ) = 3 2 x 4 − 2 x 3 − 6 x 2 + 8 to evaluate the value of relative minima. h ( − 1 ) = 3 2 ( − 1 ) 4 − 2 ( − 1 ) 3 − 6 ( − 1 ) 2 + 8 = 3 2 + 2 − 6 + 8 = 4.5 Substitute 2 for x in h ( x ) = 3 2 x 4 − 2 x 3 − 6 x 2 + 8 to evaluate the value of relative minima

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Author: Tan

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Chapter 4.3, Problem 43E

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Chapter 4.3, Problem 42E

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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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Solution Summary: The author illustrates how the function is a polynomial function. Substitute 0 for x to evaluate the y -intercept.

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