Question

The intervals of concavity for $f\left(x\right)=\frac{x}{(x-1)}$ is :

• A. Concave up for $x\in (1,\infty )$
• B. Concave down for $x\in (-\infty ,1)$
• C. Concave down for $(0,1)\cup (1,\infty )$
• D. Concave up for $(-\infty ,1)$

Answer 1

Answer: (A), (B)

Concave down for $x\in (-\infty ,1)$

$f\left(x\right)=\frac{x}{(x-1)}$
$f^{\prime }\left(x\right)=\frac{-1}{(x-1{)}^2}$
$f^{\prime \prime }\left(x\right)=\frac{2}{(x-1{)}^3}$
$\therefore f^{\prime \prime }\left(x\right)>0\Rightarrow x\in (1,\infty )$
Concave up for $x\in (1,\infty )$
Similarly, $f^{\prime \prime }\left(x\right)<0\Rightarrow x\in (-\infty ,1)$
Concave down for $x\in (-\infty ,1)$