Question

The Intervals of concavity and convexity of $f\left(x\right)=\frac{\ln x}{x}$ are

• A. Convex :$(e^{\frac{3}{2}},\infty )$
  
  
• B. Concave : $(0,e^{\frac{3}{2}})$
• C. Convex : $(0,e^{\frac{3}{2}})$
• D. Concave :$(e^{\frac{3}{2}},\infty )$
  

Answer 1

Answer: (A), (B)

Concave : $(0,e^{\frac{3}{2}})$

$f\left(x\right)=\frac{\ln x}{x}$
$f^{\prime }\left(x\right)=\frac{1-\ln x}{x^2}$
$f^{\prime \prime }\left(x\right)=\frac{2\ln x-3}{x^3}$
$f^{\prime \prime }\left(x\right)>0\Rightarrow$ convex and $f^{\prime \prime }\left(x\right)<0\Rightarrow$ concave.
So, convex in $x\in (e^{\frac{3}{2}},\infty )$
Concave in $x\in (0,e^{\frac{3}{2}})$