If `f'(x)=varphi(x)` and `varphi'(x)=f(x)` for all x. Also, `f(3)=5` and `f'(3)=4`. Then the value of `[f(3)]^(2)-[varphi(3)]^(2)` is

Options

A. 0
B. 9
C. 41
D. None of these

Answer

To solve the problem, we need to find the value of \( f(3)^2 - \varphi(3)^2 \) given the conditions \( f'(x) = \varphi(x) \), \( \varphi'(x) = f(x) \), \( f(3) = 5 \), and \( f'(3) = 4 \).

### Step-by-Step Solution:

1. **Understanding the Functions:**
We have two functions: \( f(x) \) and \( \varphi(x) \). The derivative of \( f(x) \) is \( \varphi(x) \) and the derivative of \( \varphi(x) \) is \( f(x) \). This indicates a relationship between the two functions.

2. **Define a New Function:**
Let's define a new function \( g(x) = f(x)^2 - \varphi(x)^2 \). We want to find \( g(3) \).

3. **Differentiate \( g(x) \):**
Using the product rule, we differentiate \( g(x) \):
\[ g'(x) = 2f(x)f'(x) - 2\varphi(x)\varphi'(x) \]
Substituting \( f'(x) = \varphi(x) \) and \( \varphi'(x) = f(x) \):
\[ g'(x) = 2f(x)\varphi(x) - 2\varphi(x)f(x) = 0 \]
This shows that \( g'(x) = 0 \), meaning \( g(x) \) is a constant function.

4. **Evaluate \( g(3) \):**
Since \( g(x) \) is constant, we can evaluate it at any point. We will evaluate it at \( x = 3 \):
\[ g(3) = f(3)^2 - \varphi(3)^2 \]
We need to find \( \varphi(3) \). Since \( \varphi(x) = f'(x) \), we have:
\[ \varphi(3) = f'(3) = 4 \]

5. **Substituting Known Values:**
Now we can substitute the known values into \( g(3) \):
\[ g(3) = f(3)^2 - \varphi(3)^2 = 5^2 - 4^2 \]
Calculating this gives:
\[ g(3) = 25 - 16 = 9 \]

### Final Answer:
Thus, the value of \( f(3)^2 - \varphi(3)^2 \) is \( \boxed{9} \).