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Let f(x) be a function defined for x > 0 such that f(x) = x^3 x + x^2 f'(1) + x f''(1) . The value of f'(e) is

Options

  1. A(2e - 1)^2
  2. B3e^2 - 2e + 1
  3. Ce^2 - 4e + 1
  4. De^2 - 2e

Correct answer

A. (2e - 1)^2

Step-by-step solution

Let A = f'(1) and B = f''(1) . The function can be written as: f(x) = x^3 x + Ax^2 + Bx Differentiating f(x) with respect to x using the product rule: f'(x) = 3x^2 x + x^3 ( 1 x ) + 2Ax + B f'(x) = 3x^2 x + x^2 + 2Ax + B Differentiating again to find f''(x) : f''(x) = 6x x + 3x^2 ( 1 x ) + 2x + 2A f''(x) = 6x x + 3x + 2x + 2A = 6x x + 5x + 2A Evaluate the derivatives at x = 1 to form a system of equations: f'(1) = A 3(1)^2 1 + (1)^2 + 2A(1) + B = A 0 + 1 + 2A + B = A A + B = -1 f''(1) = B 6(1) 1 + 5(1) + 2A = B 0 +

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