Properties of Differentiable Composite and Inverse Functions

To determine the correct options, let us analyze the given functions and their relationships.

We are given $f(x) = x^3 + 3x + 2$. The derivative of $f(x)$ is: $f'(x) = 3x^2 + 3$

Since $f'(x) \ge 3 > 0$ for all $x \in \mathbb{R}$, $f(x)$ is strictly increasing and bijective from $\mathbb{R} \to \mathbb{R}$.

Given $g(f(x)) = x$ for all $x \in \mathbb{R}$, $g(x)$ is the inverse function of $f(x)$, i.e., $g = f^{-1}$. Thus, we also have: $f(g(x)) = x \quad \text{for all } x \in \mathbb{R}$

Next, we are given $h(g(g(x))) = x$ for all $x \in \mathbb{R}$. Let $y = g(g(x))$. Then $h(y) = x$. Applying $f$ to both sides of $y = g(g(x))$: $f(y) = f(g(g(x))) = g(x)$ Applying $f$ again: $f(f(y)) = f(g(x)) = x$ Since $h(y) = x$, we get: $h(y) = f(f(y)) \quad \implies \quad h(x) = f(f(x)) \quad \text{for all } x \in \mathbb{R}$

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Checking Option (A):

We differentiate $g(f(x)) = x$ with respect to $x$ using the chain rule: $g'(f(x)) \cdot f'(x) = 1 \implies g'(f(x)) = \frac{1}{f'(x)}$

To find $g'(2)$, we set $f(x) = 2$: $x^3 + 3x + 2 = 2 \implies x(x^2 + 3) = 0 \implies x = 0$

So, $f(0) = 2$. Thus: $g'(2) = g'(f(0)) = \frac{1}{f'(0)}$

Since $f'(0) = 3(0)^2 + 3 = 3$: $g'(2) = \frac{1}{3}$

Hence, Option (A) is incorrect.

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Checking Option (B):

Using $h(x) = f(f(x))$, we differentiate with respect to $x$: $h'(x) = f'(f(x)) \cdot f'(x)$

For $x = 1$: $f(1) = 1^3 + 3(1) + 2 = 6$ $f'(1) = 3(1)^2 + 3 = 6$ $f'(f(1)) = f'(6) = 3(6)^2 + 3 = 108 + 3 = 111$

Therefore: $h'(1) = f'(f(1)) \cdot f'(1) = 111 \cdot 6 = 666$

Hence, Option (B) is correct.

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Checking Option (C):

Using $h(x) = f(f(x))$ for $x = 0$: $f(0) = 0^3 + 3(0) + 2 = 2$ $h(0) = f(f(0)) = f(2)$ $f(2) = 2^3 + 3(2) + 2 = 8 + 6 + 2 = 16$

Thus, $h(0) = 16$.

Hence, Option (C) is correct.

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Checking Option (D):

Using $h(x) = f(f(x))$, we evaluate $h(g(3))$: $h(g(3)) = f(f(g(3)))$

Since $f(g(x)) = x$ for all $x \in \mathbb{R}$, we have $f(g(3)) = 3$. Therefore: $h(g(3)) = f(3) = 3^3 + 3(3) + 2 = 27 + 9 + 2 = 38$

Hence, Option (D) is incorrect.

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Conclusion:

The correct options are B and C.