Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \right)} \right) = x$$ and $$h\left( {g\left( {g\left( x \right)} \right)} \right) = x$$ for all $$x \in R$$. Then

Let $$F:R \to R$$ be a thrice differentiable function. Suppose that
$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F'\left( x \right) < 0$$ for all $$x \in \left( {{1 \over 2},3} \right).$$ Let $$f\left( x \right) = xF\left( x \right)$$ for all $$x \in R.$$

The correct statement(s) is (are)