MHT CET202521 Apr 2025Morning ShiftMathematicsVector AlgebraActual
The maximum value and minimum value of the volume of the parallelopiped having coterminous edges i +x j + k , j +x k and x i + k are respectively
Options
- A1 3 3 +1, -1 3 3 +1
- B2 3 3 +1, -2 3 3 +1
- C1 3 +1, -1 3 +1
- D2 3 +1, -2 3 +1
Correct answer
B. 2 3 3 +1, -2 3 3 +1
Step-by-step solution
The volume V of the parallelepiped with coterminous edges a = i +x j + k , b = j +x k , and c = x i + k is given by the scalar triple product: V(x) = |[ a b c ]| = | vmatrix 1 & x & 1 0 & 1 & x x & 0 & 1 vmatrix | Evaluating the determinant yields V(x) = |x^3 - x + 1| . To find the extrema of V(x) , consider f(x) = x^3 - x + 1 since both critical values are positive. The derivative f'(x) = 3x^2 - 1 vanishes at x = 1 3 . The second derivative f''(x) = 6x shows f(x) has a local minimum at x = 1 3 and a local maximum