If x₀ is the point of local minima of f (x)= a ( b c ) where a =x i -2 j +3 k , b =-2 i +x j - k , c =7 i -2 j +x k , then value of a b at x=x₀ is

Step-by-step solution

aligned & f (x)= a ( b c ) & = | array ccc x & -2 & 3 -2 & x & -1 7 & -2 & x array | & =x (x^2-2 )+2(-2 x+7)+3(4-7 x) & f (x)=x^3-27 x+26 & Now, f ^ (x)=0 & 3 x^2-27=0 & x^2=9 & x= 3 & f ^ (x)=6 x & f ^ (x)=18 0 & f (x) has local minimum at x=3 . aligned aligned a & =3 i -2 j +3 k b & =-2 i +3 j - k a b & =(3 i -2 j +3 k ) (-2 i +3 j - k ) & =3(-2)+(-2)(3)+3(-1) & =-15 aligned