If the volume of the parallelepiped formed by i+ j+ k, j+ k, i+ k is minimum, then is:

Question

If the volume of the parallelepiped formed by $\hat i+\lambda\hat j+\hat k$, $\hat j+\lambda\hat k$, $\lambda\hat i+\hat k$ is minimum, then $\lambda$ is:

Accepted Answer

$$V = \begin{vmatrix}1&\lambda&1\0&1&\lambda\ \lambda&0&1\end{vmatrix} = |\lambda^3-\lambda+1|$$

Let $f(\lambda)=\lambda^3-\lambda+1$. $f'(\lambda)=3\lambda^2-1=0 \Rightarrow \lambda=\pm\dfrac1{\sqrt3}$. $f''(\lambda)=6\lambda$; $f''\left(\dfrac1{\sqrt3}\right)>0$, so $\lambda=\dfrac1{\sqrt3}$ gives a local minimum of $f$ (and hence of $|f|$ near that point, per the source).

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