The function `f: RvecR` satisfies `f(x^2)dotf^(x)=f^(prime)(x)dotf^(prime)(x^2)` for all real `xdot` Given that `f(1)=1` and `f^(1)=8` , then the value of `f^(prime)(1)+f^(1)` is `2` b. `4` c. `6` d. 8

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem, we will analyze the given functional equation and the conditions provided.

### Given:
The function \( f: \mathbb{R} \to \mathbb{R} \) satisfies the equation:
\[ f(x^2) \cdot f'(x) = f'(x) \cdot f'(x^2) \]
for all real \( x \).

We also have:
- \( f(1) = 1 \)
- \( f'(1) = 8 \)

We need to find the value of \( f'(1) + f(1) \).

### Step 1: Analyze the functional equation

From the functional equation:
\[ f(x^2) \cdot f'(x) = f'(x) \cdot f'(x^2) \]

We can simplify this by dividing both sides by \( f'(x) \) (assuming \( f'(x) \neq 0 \)):
\[ f(x^2) = f'(x^2) \]

### Step 2: Substitute \( x = 1 \)

Now, let's substitute \( x = 1 \) into the simplified equation:
\[ f(1^2) = f'(1^2) \]
This gives us:
\[ f(1) = f'(1) \]

### Step 3: Use the given values

From the problem, we know:
- \( f(1) = 1 \)
- \( f'(1) = 8 \)

Substituting these values into the equation:
\[ 1 = 8 \]
This is a contradiction, indicating that we cannot divide by \( f'(x) \) at \( x = 1 \). Therefore, we need to analyze the case when \( f'(x) = 0 \).

### Step 4: Revisit the functional equation

Since \( f'(1) \neq 0 \), we can conclude that the functional equation holds for other values of \( x \). Let's explore what happens when we set \( x = 0 \):
\[ f(0^2) \cdot f'(0) = f'(0) \cdot f'(0^2) \]
This simplifies to:
\[ f(0) \cdot f'(0) = f'(0) \cdot f'(0) \]

If \( f'(0) \neq 0 \), we can divide both sides by \( f'(0) \):
\[ f(0) = f'(0) \]

### Step 5: Calculate \( f'(1) + f(1) \)

Now, we need to find \( f'(1) + f(1) \):
\[ f'(1) + f(1) = 8 + 1 = 9 \]

However, this does not match the options provided. Let's check the calculations again.

### Final Calculation

Given \( f(1) = 1 \) and \( f'(1) = 8 \):
\[ f'(1) + f(1) = 8 + 1 = 9 \]

### Conclusion

It seems there is a misunderstanding in the problem statement or the options provided. Based on the calculations, the value of \( f'(1) + f(1) \) is \( 9 \).