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Let f(x) and g(x) be polynomials defined as f(x) = x^2 + x g'(1) + g''(2) and g(x) = x^3 + x^2 f'(1) + x f''(2) for all x R . The value of f(10) + g(10) is equal to

Correct answer

328

Step-by-step solution

Given the functions: f(x) = x^2 + x g'(1) + g''(2) g(x) = x^3 + x^2 f'(1) + x f''(2) Differentiating f(x) with respect to x : f'(x) = 2x + g'(1) f''(x) = 2 Thus, f''(2) = 2 . Substitute f''(2) = 2 into g(x) : g(x) = x^3 + x^2 f'(1) + 2x Differentiating g(x) with respect to x : g'(x) = 3x^2 + 2x f'(1) + 2 g''(x) = 6x + 2f'(1) Evaluate g'(x) at x = 1 and g''(x) at x = 2 : g'(1) = 3(1)^2 + 2(1)f'(1) + 2 = 5 + 2f'(1) g''(2) = 6(2) + 2f'(1) = 12 + 2f'(1) Now, evaluate f'(x) = 2x + g'(1) at x = 1 : f'(1) = 2(1) + g'(1) S

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