Let f(x) be a polynomial function defined for all x R such that f(x) = x^3 + x^2 f'(-1) + x f''(1) + f'''(2) . The local minimum value of f(x) is

Step-by-step solution

Let a = f'(-1) , b = f''(1) , and c = f'''(2) . The function can be written as: f(x) = x^3 + ax^2 + bx + c Differentiating f(x) successively with respect to x , we get: f'(x) = 3x^2 + 2ax + b f''(x) = 6x + 2a f'''(x) = 6 Now, substitute the respective values of x to form a system of equations: f'(-1) = a 3(-1)^2 + 2a(-1) + b = a 3 - 2a + b = a 3a - b = 3 f''(1) = b 6(1) + 2a = b -2a + b = 6 f'''(2) = c c = 6 Adding the first two equations: (3a - b) + (-2a + b) = 3 + 6 a = 9 Substitute a = 9 into -2a + b = 6 : -18 +