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 $f(x)=x^{3}+3x+2$, $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in\mathbb{R}$. Then

$f(x) = x^3 + 3x + 2$ $g(f(x)) = x, \quad h(g(g(x))) = x$ $\text{Which of the four statements is/are correct?}$ $f'(x) = 3x^2 + 3 > 0 \Rightarrow f \text{ is invertible},\ g = f^{-1}$ $g(2) = 0 \text{ since } f(0) = 2$ $g'(2) = \dfrac{1}{f'(0)} = \dfrac{1}{3} \neq \dfrac{1}{15} \Rightarrow (1) \text{ is false}$ $h \circ g \circ g = \mathrm{id} \Rightarrow h = (g \circ g)^{-1} = f \circ f$ $h'(x) = f'(f(x))\,f'(x)$ $h'(1) = f'(6)\cdot f'(1) = 111 \cdot 6 = 666 \Rightarrow (2) \text{ is true}$ $h(0) = f(f(0)) = f(2) = 16 \Rightarrow (3) \text{ is true}$ $h(g(3)) = f(f(g(3))) = f(3) = 38 \neq 36 \Rightarrow (4) \text{ is false}$ $\boxed{\text{Correct options: } (2) \text{ and } (3)}$