Chapter 4 | Applications of Derivatives
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Figure 4.23 This function is continuous and differentiable over [−2, 0], f ′( c ) =0 when c =−1.
b. As in part a. f is a polynomial and therefore is continuous and differentiable everywhere. Also, f (−2) =0= f (2). That said, f satisfies the criteria of Rolle’s theorem. Differentiating, we find that f ′( x ) =3 x 2 −4. Therefore, f ′( c ) =0 when x =± 2 3 . Both points are in the interval [−2, 2], and, therefore, both points satisfy the conclusion of Rolle’s theorem as shown in the following graph.
Figure 4.24 For this polynomial over [−2, 2], f ′( c ) =0 at x =±2/ 3.
4.14 Verify that the function f ( x ) =2 x 2 −8 x +6 defined over the interval [1, 3] satisfies the conditions of Rolle’s theorem. Find all points c guaranteed by Rolle’s theorem.
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