A nonuniform rod $OM$  of length $\mathrm{l}$ has linear mass density that varies with the distance $x$ from left end of the rod according to $\lambda ={\lambda }_0\left(\frac{x^3}{L^3}\right)$; Where ${\lambda }_0$  is constant, is kept along $x$-axis and is rotating about an axis $AB$, which is perpendicular to the rod as shown in the figure. What is the value of $x$  so that moment of inertia of rod about axis $A\mathrm{B }\left({\text{I}}_{AB}\right)$  is minimum?

 

• A

  $\frac{7\mathrm{\ell }}{15}$
• B

  $\frac{2\ell }{5}$
• C

  $\frac{8\mathcal{l}}{15}$
• Correct answer
  D

  $\frac{4\ell }{5}$

Hence, the correct answer is option: 'D'.