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JEE MainMathematicsDifferentiation

Let f(x) = x^3 + vmatrix x^2 & x & 1 f'(1) & f''(2) & f'''(3) 1 & 1 & 0 vmatrix , x R . Then the value of f(10) is equal to

Correct answer

457

Step-by-step solution

Given f(x) = x^3 + vmatrix x^2 & x & 1 f'(1) & f''(2) & f'''(3) 1 & 1 & 0 vmatrix Expanding the determinant along the first row, we get: f(x) = x^3 + x^2(0 - f'''(3)) - x(0 - f'''(3)) + 1(f'(1) - f''(2)) f(x) = x^3 - f'''(3)x^2 + f'''(3)x + f'(1) - f''(2) Differentiating with respect to x successively: f'(x) = 3x^2 - 2f'''(3)x + f'''(3) f''(x) = 6x - 2f'''(3) f'''(x) = 6 From this, we have f'''(3) = 6 . Substitute f'''(3) = 6 into the expression for f''(x) : f''(x) = 6x - 12 At x = 2 , f''(2) = 12 - 12 = 0 . Substi

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