`f'(x) = g(x) and g'(x) =-f(x)` for all real x and `f(5)=2=f'(5) then f^2 (10) + g^2 (10)` is

Options

A. 2
B. 4
C. 8
D. None of these

Answer

To solve the problem, we need to find the value of \( f^2(10) + g^2(10) \) given the relationships \( f'(x) = g(x) \) and \( g'(x) = -f(x) \), along with the initial conditions \( f(5) = 2 \) and \( f'(5) = 2 \).

### Step-by-step Solution:

1. **Define a new function**:
Let \( h(x) = f^2(x) + g^2(x) \).

2. **Differentiate \( h(x) \)**:
Using the chain rule, we differentiate \( h(x) \):
\[ h'(x) = 2f(x)f'(x) + 2g(x)g'(x) \]

3. **Substitute \( f'(x) \) and \( g'(x) \)**:
We know from the problem that \( f'(x) = g(x) \) and \( g'(x) = -f(x) \). Substitute these into the derivative:
\[ h'(x) = 2f(x)g(x) + 2g(x)(-f(x)) \]
Simplifying this, we find:
\[ h'(x) = 2f(x)g(x) - 2f(x)g(x) = 0 \]

4. **Conclusion about \( h(x) \)**:
Since \( h'(x) = 0 \), \( h(x) \) is a constant function. This means that \( h(x) \) does not change with \( x \).

5. **Evaluate \( h(5) \)**:
We need to find \( h(5) \):
\[ h(5) = f^2(5) + g^2(5) \]
We know \( f(5) = 2 \). Now we need to find \( g(5) \):
\[ g(5) = f'(5) = 2 \]
Therefore:
\[ h(5) = 2^2 + 2^2 = 4 + 4 = 8 \]

6. **Since \( h(x) \) is constant**:
We conclude that:
\[ h(10) = h(5) = 8 \]
Thus:
\[ f^2(10) + g^2(10) = 8 \]

### Final Answer:
\[ f^2(10) + g^2(10) = 8 \]