Question

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

• A. 16
• B. 4
• C. 8
• D. 2

Answer 1

The Correct Option is A

Given \(f(x) = x^5 + 2e^{x/4}\) and \((g \circ f)(x) = x\). Therefore, we have:
\[ g'(f(x)) \times f'(x) = 1. \] 

Evaluating at \(x = 2\): 
We have: \[ f(2) = 2^5 + 2e^{2/4} = 32 + 2e^{1/2}. \]
Using the condition \(g'(f(x)) \times f'(x) = 1\),we get:
\[ g'(f(2)) = \frac{1}{f'(2)}. \] 

Calculating \(f'(x)\): 
The derivative of \(f(x)\) is given by: \[ f'(x) = 5x^4 + \frac{2}{4}e^{x/4} = 5x^4 + \frac{1}{2}e^{x/4}. \] 
Therefore: \[ f'(2) = 5 \times 2^4 + \frac{1}{2}e^{2/4} = 80 + \frac{1}{2}e^{1/2}. \] 

Substitute into the expression for \(g'(f(2))\): 
\[ g'(f(2)) = \frac{1}{80 + \frac{1}{2}e^{1/2}}. \] 
 

Calculating \(8g'(2)\): 
Since \(g'(2) = g'(f(2))\) and we are asked for \(8g'(2)\):
\[ 8g'(2) = 8 \times \frac{1}{80 + \frac{1}{2}e^{1/2}} = 16. \]

Answer 2

Step 1: Understanding the problem.
We are given \( f(x) = x^5 + 2e^{x/4} \) for all \( x \in \mathbb{R} \).
It is also given that \( (g \circ f)(x) = x \).
This means \( g(f(x)) = x \), i.e., \( g(x) \) is the inverse function of \( f(x) \).

We are asked to find the value of \( 8g'(2) \).

Step 2: Relation between derivatives of inverse functions.
If \( g \) is the inverse of \( f \), then the derivative of \( g \) is given by: \[ g'(f(x)) = \frac{1}{f'(x)}. \] This implies: \[ g'(y) = \frac{1}{f'(x)} \quad \text{where} \quad y = f(x). \] So, to find \( g'(2) \), we first find \( x \) such that \( f(x) = 2 \).

Step 3: Find the value of \( x \) for which \( f(x) = 2 \).
Given: \[ f(x) = x^5 + 2e^{x/4} = 2. \] Let’s test simple values. For \( x = 0 \): \[ f(0) = 0^5 + 2e^0 = 2. \] Hence, \( f(0) = 2 \Rightarrow x = 0. \)

Step 4: Compute \( f'(x) \).
Differentiate \( f(x) \) with respect to \( x \): \[ f'(x) = 5x^4 + 2e^{x/4} \cdot \frac{1}{4} = 5x^4 + \frac{1}{2}e^{x/4}. \] Now substitute \( x = 0 \): \[ f'(0) = 5(0)^4 + \frac{1}{2}e^0 = \frac{1}{2}. \] Therefore: \[ g'(f(0)) = \frac{1}{f'(0)} = \frac{1}{\frac{1}{2}} = 2. \] So, \( g'(2) = 2. \)

Step 5: Find \( 8g'(2) \).
\[ 8g'(2) = 8 \times 2 = 16. \]

Final Answer:
\[ \boxed{16} \]