If and is a differentiable bijective function

MATHEMATICS · Multi correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

f:\mathbb{R}\to\mathbb{R}

and

f(x)

is a differentiable bijective function, then which of the following may be true? (1)

(f(x)-x)f''(x)<0

, for all

x\in\mathbb{R}

(2)

(f(x)-x)f''(x)>0

, for all

x\in\mathbb{R}

(3) If

(f(x)-x)f''(x)>0

, then

f(x)=f^{-1}(x)

has no solution. (4) If

(f(x)-x)f''(x)>0

, then

f(x)=f^{-1}(x)

has at least one real solution.