Science:Math Exam Resources/Courses/MATH110/April 2016/Question 08 (b)/Solution 1

A linear apprxoimation will be an underestimate if the function $f(x)=\frac{1}{x}$ is concave up between 4 and 4.16, and will be an overestimate if $f(x)$ is concave down on this interval. We computed in part (a) that $f^{\prime }(x)=-\frac{1}{x^2}$. We can use this to compute $f^{″}(x)$:

$f^{″}(x)=\frac{2}{x^3}.$

This is positive between the values $x=4$ and $x=4.16$. Therefore, the function is concave up and the estimate is an underestimate.

Answer: $\frac{1}{4.16}>0.24.$