Mathematics

Mathematics

Problem

The function hh is differentiable, and its tangent line at x=4x=4 is used to estimate nearby values. The tangent-line formula gives htan(4.05)=7.26h_{\text{tan}}(4.05)=7.26, while the actual value is h(4.05)=7.31h(4.05)=7.31. Which of the following statements about h′′(4)h''(4) is most consistent with this information? a) h′′(4)>0h''(4)>0 because the tangent-line estimate is an overestimate. b) h′′(4)<0h''(4)<0 because the tangent-line estimate is an underestimate. c) h′′(4)>0h''(4)>0 because the tangent-line estimate is an underestimate. d) h′′(4)=0h''(4)=0 because the estimate and actual value differ by only 0.050.05.

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