Let `f(x) = x^5 + 2e^(x/4) AA x in R`. Consider a function of (gof) `(x) = x, AA x in R` then the value of 8g' (2) is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the value of \(8g'(2)\) given the function \(f(x) = x^5 + 2e^{x/4}\) and the composition \(g(f(x)) = x\).

### Step-by-Step Solution:

1. **Differentiate the Composition**:
We start with the equation:
\[ g(f(x)) = x \]
We differentiate both sides with respect to \(x\):
\[ g'(f(x)) \cdot f'(x) = 1 \]
From this, we can express \(g'(f(x))\) as:
\[ g'(f(x)) = \frac{1}{f'(x)} \]

2. **Find \(g'(2)\)**:
To find \(g'(2)\), we need to find \(x\) such that \(f(x) = 2\). We set:
\[ f(a) = a^5 + 2e^{a/4} = 2 \]
We need to solve this equation for \(a\).

3. **Check for \(a = 0\)**:
Let's check if \(a = 0\) satisfies the equation:
\[ f(0) = 0^5 + 2e^{0/4} = 0 + 2 \cdot 1 = 2 \]
Thus, \(f(0) = 2\), so \(a = 0\) is a solution.

4. **Substitute \(a = 0\) into \(g'\)**:
Since \(f(0) = 2\), we can substitute \(a = 0\) into our expression for \(g'\):
\[ g'(2) = g'(f(0)) = \frac{1}{f'(0)} \]

5. **Find \(f'(x)\)**:
Now, we need to compute \(f'(x)\):
\[ f'(x) = 5x^4 + 2 \cdot \frac{1}{4} e^{x/4} = 5x^4 + \frac{1}{2} e^{x/4} \]
Now, substituting \(x = 0\):
\[ f'(0) = 5 \cdot 0^4 + \frac{1}{2} e^{0/4} = 0 + \frac{1}{2} \cdot 1 = \frac{1}{2} \]

6. **Calculate \(g'(2)\)**:
Now we can find \(g'(2)\):
\[ g'(2) = \frac{1}{f'(0)} = \frac{1}{\frac{1}{2}} = 2 \]

7. **Calculate \(8g'(2)\)**:
Finally, we compute:
\[ 8g'(2) = 8 \cdot 2 = 16 \]

### Final Answer:
The value of \(8g'(2)\) is \(16\).