Question

If the volume of the parallelopiped with coterminal of edges $\hat{i}+a\hat{j}+\hat{k}$, $\hat{j}+a\hat{k},$ $a\hat{i}+\hat{k}$,is minimum,then $a=$

• A. $3$
• B. $\frac{1}{\sqrt{3}}$
• C. $\frac{1}{3}$
• D. $\sqrt{3}$

Answer 1

Answer: (B)

$\frac{1}{\sqrt{3}}$

Volume of a parallelopiped$=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})$
$=\left[\begin{array}{ccc}1& a& 1\\ 0& 1& a\\ a& 0& 1\end{array}\right]$
$\Rightarrow f\left(a\right)=1+a^3-a$(on evaluation)
$\Rightarrow f^{\prime }\left(a\right)=3a^2-1$
$\Rightarrow f^{\prime }\left(a\right)=0$ for $a=\pm \frac{1}{\sqrt{3}}$
$\Rightarrow f^{\prime \prime }\left(a\right)=6a=6\times \frac{1}{\sqrt{3}}=\frac{6}{\sqrt{3}}>0$
$\therefore f\left(a\right)$ is minimum for $a=\frac{1}{\sqrt{3}}$