Let x0 be the point of local maxima of f(x)=a⋅(b×c), where a=xi^−2j^+3k^, b=−2i^+xj^−k^, c=7i^−2j^+xk^. Then a⋅b+b⋅c+c⋅a at x=x0 is:
Solutions · 1
Official solution
\[f(x) = \begin{vmatrix}x&-2&3\\-2&x&-1\\7&-2&x\end{vmatrix} = x^3-27x+26\]
f′(x)=3x2−27=0⇒x=±3. Since f′′(x)=6x, f′′(−3)<0: local maximum at x0=−3.
\[\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a = 3x-13\ \text{(as a function of }x\text{, per the source)}\]
At x=−3: 3(−3)−13=−22.