Science:Math Exam Resources/Courses/MATH100/December 2014/Question 08 (f)/Solution 1

The second derivative is defined everywhere on the domain. The only zero occurs at

$\begin{array}{rl}3\log (x)& =8\\ x& =e^{8/3}\end{array}$

So we examine:

$\begin{array}{rlr}(0,e^{8/3}):f^{″}(x)& =\frac{neg}{pos}=neg& \text{concave down}\\ (e^{8/3},\mathrm{\infty }):f^{″}(x)& =\frac{pos}{pos}=pos& \text{concave up}\end{array}$

So, the only inflection point is at $(e^{\frac{8}{3}},f(e^{\frac{8}{3}}))=(e^{\frac{8}{3}},\frac{8}{3e^{4/3}})$.