If `f'(x)=g(x)` and `g'(x)=-f(x)` for all x and `f(2) =4 =f'(2)` then `f^(2)(16) + g^(2)(16)` is:

Options

A. 16
B. 32
C. 64
D. none

Answer

To solve the problem, we need to analyze the given information and use the relationships between the functions \( f(x) \) and \( g(x) \).

Given:
1. \( f'(x) = g(x) \)
2. \( g'(x) = -f(x) \)
3. \( f(2) = 4 \)
4. \( f'(2) = 4 \)

We need to find \( f^{(2)}(16) + g^{(2)}(16) \).

### Step 1: Find the second derivatives \( f^{(2)}(x) \) and \( g^{(2)}(x) \)

From the first derivative:
- \( f'(x) = g(x) \)

Taking the derivative of both sides:
- \( f^{(2)}(x) = g'(x) \)

Using the second given relationship:
- \( g'(x) = -f(x) \)

Thus, we have:
- \( f^{(2)}(x) = -f(x) \)

### Step 2: Find \( g^{(2)}(x) \)

From \( g'(x) = -f(x) \), we can differentiate again:
- \( g^{(2)}(x) = -f'(x) \)

Using the first relationship:
- \( f'(x) = g(x) \)

Thus, we have:
- \( g^{(2)}(x) = -g(x) \)

### Step 3: Evaluate at \( x = 16 \)

Now we need to evaluate \( f^{(2)}(16) + g^{(2)}(16) \):
- \( f^{(2)}(16) = -f(16) \)
- \( g^{(2)}(16) = -g(16) \)

So:
\[ f^{(2)}(16) + g^{(2)}(16) = -f(16) - g(16) = -(f(16) + g(16)) \]

### Step 4: Find \( f(16) \) and \( g(16) \)

To find \( f(16) \) and \( g(16) \), we can use the information we have. Since \( f(2) = 4 \) and \( f'(2) = 4 \), we can assume a form for \( f(x) \) and \( g(x) \).

Assuming \( f(x) \) and \( g(x) \) are sinusoidal functions, we can express them as:
- \( f(x) = A \cos(x - \phi) \)
- \( g(x) = A \sin(x - \phi) \)

From the initial conditions, we can find the constants \( A \) and \( \phi \).

However, we can also use the fact that \( f^{(2)}(x) + g^{(2)}(x) = 0 \) implies that \( f^{(2)}(x) + g^{(2)}(x) \) is a constant function. Therefore, we can evaluate it at any point.

### Step 5: Calculate the final result

Since we know:
- \( f^{(2)}(x) + g^{(2)}(x) = 0 \)

At \( x = 2 \):
\[ f^{(2)}(2) + g^{(2)}(2) = -f(2) - g(2) = -4 - 4 = -8 \]

Since the second derivatives are constant, we can conclude:
\[ f^{(2)}(16) + g^{(2)}(16) = -8 \]

Thus, the final answer is:
\[ \boxed{-8} \]