Science:Math Exam Resources/Courses/MATH100/December 2011/Question 05 (d)/Solution 1

From part (c), we know that

${\begin{aligned}f''(x)&={\frac {6(x-4)}{25x^{9/5}}}\end{aligned}}$

Notice that the potential inflection points are at x=0,4, where $f''(x)$ is either zero or undefined. Doing a sign chart for this function yields the following. When x < 0 we have that $f''(x)>0$. When $0<x<4$ we have that $f''(x)<0$. When $x>4$ we have that $f''(x)>0$. Hence, the function is concave up on $(-\infty ,0)\cup (4,\infty )$.

Note: Since the sign of $f''(x)$ changes at both points, both $x=0$ and $x=4$ are inflection points. Even though $f''(x)$ is not defined at $x=0$.