Let `f(x)` be differentiable for real `x` such that `f^(prime)(x)>0on(-oo,-4),` `f^(prime)(x)<0on(-4,6),` `f^(prime)(x)>0on(6,oo),` If `g(x)=f(10-2x),` then the value of `g^(prime)(2)` is a. 1 b. 2 c. 0 d. 4

Options

A. 1
B. 2
C. 0
D. 4

Explanation

To solve the problem, we need to find the value of \( g'(2) \) given that \( g(x) = f(10 - 2x) \) and the behavior of the derivative \( f'(x) \) over different intervals.

### Step-by-Step Solution:

1. **Identify the function \( g(x) \)**:
\[ g(x) = f(10 - 2x) \]

2. **Differentiate \( g(x) \)** using the chain rule:
\[ g'(x) = f'(10 - 2x) \cdot (-2) \]

3. **Evaluate \( g'(2) \)**:
\[ g'(2) = -2 \cdot f'(10 - 2 \cdot 2) = -2 \cdot f'(10 - 4) = -2 \cdot f'(6) \]

4. **Determine the value of \( f'(6) \)**:
From the problem statement, we know:
- \( f'(x) > 0 \) on \( (-\infty, -4) \)
- \( f'(x) < 0 \) on \( (-4, 6) \)
- \( f'(x) > 0 \) on \( (6, \infty) \)

Since \( 6 \) falls in the interval \( (-4, 6) \), we have:
\[ f'(6) = 0 \]

5. **Substitute \( f'(6) \) back into the equation for \( g'(2) \)**:
\[ g'(2) = -2 \cdot f'(6) = -2 \cdot 0 = 0 \]

### Conclusion:
The value of \( g'(2) \) is \( 0 \).

### Final Answer:
The correct option is **c. 0**.