Explanation Given Information: The provided function is g ( t ) = 3 t 4 − 16 t 3 + 24 t 2 + 1 and the domain is ( − ∞ , + ∞ ) . Calculation: Consider the provided function, g ( t ) = 3 t 4 − 16 t 3 + 24 t 2 + 1 Differentiate both sides of the above equation with respect to t , g ′ ( t ) = 12 t 3 − 48 t 2 + 48 t Locating stationary points: Stationary Points: Points in the interior of the domain where the derivative of the function is zero. To locate stationary points set g ′ ( t ) = 0 and solve for t . So, g ′ ( t ) = 0 12 t 3 − 48 t 2 + 48 t = 0 Take 12 t common to get: 12 t ( t 2 − 4 t + 4 ) = 0 Thus, 12 t = 0 t = 0 Or t 2 − 4 t + 4 = 0 ( t − 2 ) 2 = 0 Take square root both sides of the above equation to get: ( t − 2 ) 2 = 0 t − 2 = 0 Add 2 to both sides of the equation above to get: t − 2 + 2 = 2 t = 2 There are two solutions t = 0 and t = 2 and both are in the domain. These are the x − c o o r d i n a t e s of stationary extrema. Locating Singular points: Singular Points: Points in the interior of the domain where the derivative of the function is not defined. To locate singular points, find the interior points t where g ′ ( t ) is not defined but g ( t ) is defined. Since g ′ ( t ) is defined for all t in the interior of the domain, so there are no singular points. Locating end points: Since ( − ∞ , + ∞ ) is the domain so no end points exist

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ISBN: 8220103612005

Author: Costenoble

Publisher: CENGAGE L

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Chapter 12.1, Problem 24E

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To calculate: The exact location of the relative and absolute extrema of the function g(t)=3t4−16t3+24t2+1 with the domain (−∞,+∞).

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Chapter 12.1, Problem 23E

Chapter 12.1, Problem 25E

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Chapter 12 Solutions

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The exact location of the relative and absolute extrema of the function g ( t ) = 3 t 4 − 16 t 3 + 24 t 2 + 1 with the domain ( − ∞ , + ∞ ) .

Solution Summary: The author calculates the exact location of relative and absolute extrema of the function g(t) = 3t 4 16

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To calculate: The exact location of the relative and absolute extrema of the function g(t)=3t4−16t3+24t2+1 with the domain (−∞,+∞).

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Chapter 12 Solutions