Question

The maximum value of the function $f(x)=\frac{{\ln }^{2}x}{x}$ in $\left[\frac{1}{2},10\right]$ is

• A. $2{\ln }^2(2)$
• B. $\frac{4}{e^2}$
• C. $\frac{{\ln }^{2}10}{10}$
• D. none

Answer 1

Answer: (A)

$f^{\prime }(x)=\frac{x2\ln x\cdot \frac{1}{x}-{\ln }^{2}x}{x^2}$
$f^{\prime }(x)=\frac{\ln x}{x^2}(2-\ln x)$
$f(1)=0$ (local minima)
$f\left(e^2\right)=\frac{4}{e^2}\text{ (local maxima) }$
$\therefore \text{ Maximum occurs at }x=\frac{1}{2}$
$f\left(\frac{1}{2}\right)=2{\ln }^{2}2$