"Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :"

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Given that $(g \circ f)(x) = x$ for all $x \in \mathbf{R}$. This means $g(f(x)) = x$ for all $x \in \mathbf{R}$. Differentiating both sides with respect to $x$, we get:

$g'(f(x)) \cdot f'(x) = 1$

Now, we want to find the value of $8g'(2)$. To do this, we need to find a value of $x$ such that $f(x) = 2$. Let's solve for $x$:

$x^5 + 2e^{x/4} = 2$

By inspection, we see that $x = 0$ is a solution. Therefore, $f(0) = 2$. Now, we can substitute this into our differentiated equation:

$g'(f(0)) \cdot f'(0) = 1$

$g'(2) \cdot f'(0) = 1$

Let's find $f'(0)$:

$f'(x) = 5x^4 + \frac{1}{2}e^{x/4}$

$f'(0) = \frac{1}{2}$

Substituting this back into our equation:

$g'(2) \cdot \frac{1}{2} = 1$

$g'(2) = 2$

Finally, we can calculate $8g'(2)$:

$8g'(2) = 8 \cdot 2 = \boxed{16}$

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Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

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Let $f(x)=x^5+2 \mathrm{e}^{x / 4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8 g^{\prime}(2)$ is :

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