Question

Let $x_{0}$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c}), $ where $\quad \vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}$ $\vec{ b }=-2 \hat{ i }+ x \hat{ j }-\hat{ k }$ and $\vec{ c }=7 \hat{ i }-2 \hat{ j }+ x \hat{ k }$. Then the value of $\vec{ a } \cdot \vec{ b }+\vec{ b } \cdot \vec{ c }+\vec{ c } \cdot \vec{ a }$ at $x = x _{0}$ is :

• A. -30
• B. 14
• C. -4
• D. -22

Answer 1

Answer: (D)

$f\left(x\right) = \vec{a} .\left(\vec{b}\times\vec{c}\right) = \begin{vmatrix}x&-2&3\\ -2&x&-1\\ 7&-2&x\end{vmatrix} $
$= x^{3} - 27 x + 26$
$f'( x )=3 x ^{2}-27=0$
$ \Rightarrow x =\pm 3$
and $f''(-3)<0$
$\Rightarrow $ local maxima at $x = x _{0}=-3$
Thus, $\vec{ a }=-3 \hat{ i }-2 \hat{ j }+3 \hat{ k }$
$\vec{ b }=-2 \hat{ i }-3 \hat{ j }-\hat{ k }$
and $\vec{ c }=7 \hat{ i }-2 \hat{ j }-3 \hat{ k }$
$\Rightarrow \vec{ a } \cdot \vec{ b }+\vec{ b } \cdot \vec{ c }+\vec{ c } \cdot \vec{ a }$
$=9-5-26=-22$