Let f be a real polynomial of degree n such that f(x) = f'(x) f''(x) , for all x R . If f(0) = 0 , then 36 (f'(2) + f''(2) + ₀^2 f(x) ,dx ) is equal to:

Step-by-step solution

Let the degree of the polynomial f(x) be n . The degree of f'(x) is n-1 and the degree of f''(x) is n-2 . Since f(x) = f'(x) f''(x) , equating the degrees on both sides gives: n = (n-1) + (n-2) n = 3 Let f(x) = ax^3 + bx^2 + cx + d . Given f(0) = 0 , we get d = 0 . Thus, f(x) = ax^3 + bx^2 + cx . Differentiating f(x) with respect to x : f'(x) = 3ax^2 + 2bx + c f''(x) = 6ax + 2b Substituting these into the given equation f(x) = f'(x) f''(x) : ax^3 + bx^2 + cx = (3ax^2 + 2bx + c)(6ax + 2b) ax^3 + bx^2 + cx = 18a^2x^3