Science:Math Exam Resources/Courses/MATH104/December 2016/Question 02 (b)/Solution 1

The first derivative of m is straightforward, if the derivative is positive, then $f'>0$.

For $f''$, we use the points of inflection (3 and 5 from question). 3 is one inflection point. From 0 to 3, $f'$ goes from positive to negative, thus during that area, $f''<0$. For $t\in [3,5],f''>0$. For $t>5,f''<0$.

The part where $f',f''$ are both positive: $\left(4,5\right)$

The part where $f',f''$ are both negative: $\left(2,3\right)$

answer: $\color {blue}\left(2,3\right),\left(4,5\right)$