Let f(x) = vmatrix k+ ^2 x & ^2 x & 2x ^2 x & k+ ^2 x & 2x ^2 x & ^2 x & k+ 2x vmatrix , where k is a real number. If the maximum value of f(x) over all real x is 96 , then the value of k is equal to

Step-by-step solution

Given f(x) = vmatrix k+ ^2 x & ^2 x & 2x ^2 x & k+ ^2 x & 2x ^2 x & ^2 x & k+ 2x vmatrix Applying the column operation C₁ C₁ + C₂ + C₃ , the first column becomes: C₁ = bmatrix k + ^2 x + ^2 x + 2x ^2 x + k + ^2 x + 2x ^2 x + ^2 x + k + 2x bmatrix = bmatrix k + 1 + 2x k + 1 + 2x k + 1 + 2x bmatrix Taking (k + 1 + 2x) common from C₁ , we get: f(x) = (k + 1 + 2x) vmatrix 1 & ^2 x & 2x 1 & k+ ^2 x & 2x 1 & ^2 x & k+ 2x vmatrix Applying row operations R₂ R₂ - R₁ and R₃ R₃ - R₁ : f(x) = (k + 1 + 2x) vmatrix 1 & ^2 x & 2x