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 $f'(x)=-{\frac {1}{x^{2}}}$. 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: $\color {blue}{\frac {1}{4.16}}>0.24.$