Given the function f (x) = x2ax + B (a, B are constants) and the equation f (x) - x + 12 = 0 has two real roots X1 = 3, X2 = 4. (1) find the analytic expression of function f (x); (2) let k > 1, solve the inequality about X; f (x) ＜ (K + 1) x-k2-x | Math enthusiast | Mathematical art

Given the function f (x) = x2ax + B (a, B are constants) and the equation f (x) - x + 12 = 0 has two real roots X1 = 3, X2 = 4. (1) find the analytic expression of function f (x); (2) let k > 1, solve the inequality about X; f (x) ＜ (K + 1) x-k2-x

(1) By substituting X1 = 3 and X2 = 4 into the equation x2ax + b-X + 12 = 0, we get 93a + B = - 9164a + B = - 8 and a = - 1B = 2, so f (x) = x22-x (x ≠ 2). (2) the inequality is x22-x ＜ (k + 1) x-k2-x, which can be reduced to X2 - (K + 1) x + k2-x ＜ 0, that is, (X-2) (x-1) (x-k) ＞ 0

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