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

Step-by-step solution

The function f(x) is given by the scalar triple product of the vectors a , b , and c . f(x) = vmatrix x & -2 & 3 -2 & x & -1 7 & -2 & x vmatrix Expanding the determinant, we get: f(x) = x(x^2 - 2) - (-2)(-2x + 7) + 3(4 - 7x) f(x) = x^3 - 2x - 4x + 14 + 12 - 21x f(x) = x^3 - 27x + 26 To find the critical points, we set the first derivative to zero: f'(x) = 3x^2 - 27 = 0 x^2 = 9 x = 3 To determine the point of local maxima, we check the second derivative: f''(x) = 6x At x = 3 , f''(3) = 18 > 0 (local minima). At x =