Let be a polynomial one-one function such that f(x)f(y) + 2 = f(x) + f(y) + f(xy)

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

Let

f(x)

be a polynomial one-one function such that

f(x)f(y) + 2 = f(x) + f(y) + f(xy), \forall x, y \in R - \{0\},

f(1) \neq 1, f'(1) = 3.

Let

g(x) = \frac{x}{4}(f(x) + 3) - \int_0^x f(t) dt

, then (1)

g(x) = 0

has exactly one root for

x \in (0, 1)

(2)

g(x) = 0

has exactly two roots for

x \in (0, 1)

(3)

g(x) \neq 0, \forall x \in R - \{0\}

(4)

g(x) = 0, \forall x \in R - \{0\}