if `f(x)=3e^(x^2)` then `f'(x)-2xf(x)+1/3f(0)-f'(0)`

Options

A. 0
B. 1
C. `((7//3)e^(x^(2))`
D. `e^(x^(2))`

Answer

To solve the problem step by step, we need to calculate \( f'(x) - 2x f(x) + \frac{1}{3} f(0) - f'(0) \) given that \( f(x) = 3e^{x^2} \).

### Step 1: Find \( f'(x) \)

Given:
\[ f(x) = 3e^{x^2} \]

To find \( f'(x) \), we use the chain rule:
\[ f'(x) = \frac{d}{dx}(3e^{x^2}) = 3 \cdot e^{x^2} \cdot \frac{d}{dx}(x^2) \]
\[ \frac{d}{dx}(x^2) = 2x \]
Thus,
\[ f'(x) = 3 \cdot e^{x^2} \cdot 2x = 6xe^{x^2} \]

### Step 2: Calculate \( f(0) \)

Now, we calculate \( f(0) \):
\[ f(0) = 3e^{0^2} = 3e^0 = 3 \cdot 1 = 3 \]

### Step 3: Calculate \( f'(0) \)

Next, we calculate \( f'(0) \):
\[ f'(0) = 6 \cdot 0 \cdot e^{0^2} = 6 \cdot 0 \cdot 1 = 0 \]

### Step 4: Substitute values into the expression

Now, we substitute \( f'(x) \), \( f(x) \), \( f(0) \), and \( f'(0) \) into the expression:
\[ f'(x) - 2x f(x) + \frac{1}{3} f(0) - f'(0) \]
Substituting the values we found:
\[ = 6xe^{x^2} - 2x(3e^{x^2}) + \frac{1}{3}(3) - 0 \]
\[ = 6xe^{x^2} - 6xe^{x^2} + 1 \]

### Step 5: Simplify the expression

Now, simplify the expression:
\[ = (6xe^{x^2} - 6xe^{x^2}) + 1 = 0 + 1 = 1 \]

### Final Answer

Thus, the final answer is:
\[ \boxed{1} \]