Let `f:R rarr R` be a function such that
`f(x)=x^(3)+x^(2)f'(1)+xf''(2)+f''(3)`
What is f'(1) equal to

Options

A. `-6`
B. `-5`
C. 1
D. 0

Explanation

To solve the problem, we start with the given function:

\[ f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3) \]

We need to find \( f'(1) \).

### Step 1: Differentiate \( f(x) \)

First, we differentiate \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(x^2 f'(1)) + \frac{d}{dx}(x f''(2)) + \frac{d}{dx}(f'''(3)) \]

Calculating each term separately:

1. The derivative of \( x^3 \) is \( 3x^2 \).
2. The derivative of \( x^2 f'(1) \) is \( 2x f'(1) \) (since \( f'(1) \) is a constant).
3. The derivative of \( x f''(2) \) is \( f''(2) \) (since \( f''(2) \) is also a constant).
4. The derivative of \( f'''(3) \) is \( 0 \) (as it is a constant).

Putting it all together, we have:

\[ f'(x) = 3x^2 + 2x f'(1) + f''(2) \]

### Step 2: Evaluate \( f'(1) \)

Now we substitute \( x = 1 \) into \( f'(x) \):

\[ f'(1) = 3(1)^2 + 2(1) f'(1) + f''(2) \]

This simplifies to:

\[ f'(1) = 3 + 2 f'(1) + f''(2) \]

### Step 3: Rearrange the equation

Now, we rearrange the equation to isolate \( f'(1) \):

\[ f'(1) - 2 f'(1) = 3 + f''(2) \]

This simplifies to:

\[ -f'(1) = 3 + f''(2) \]

### Step 4: Solve for \( f'(1) \)

Multiplying both sides by -1 gives us:

\[ f'(1) = -3 - f''(2) \]

### Step 5: Find \( f''(2) \)

Next, we need to find \( f''(2) \). We differentiate \( f'(x) \) to find \( f''(x) \):

\[ f''(x) = \frac{d}{dx}(3x^2 + 2x f'(1) + f''(2)) \]

Calculating each term:

1. The derivative of \( 3x^2 \) is \( 6x \).
2. The derivative of \( 2x f'(1) \) is \( 2 f'(1) \) (as \( f'(1) \) is a constant).
3. The derivative of \( f''(2) \) is \( 0 \).

Thus, we have:

\[ f''(x) = 6x + 2 f'(1) \]

Now, substituting \( x = 2 \):

\[ f''(2) = 6(2) + 2 f'(1) = 12 + 2 f'(1) \]

### Step 6: Substitute back into the equation

Substituting \( f''(2) \) back into our equation for \( f'(1) \):

\[ f'(1) = -3 - (12 + 2 f'(1)) \]

This simplifies to:

\[ f'(1) = -3 - 12 - 2 f'(1) \]

### Step 7: Solve for \( f'(1) \)

Rearranging gives:

\[ f'(1) + 2 f'(1) = -15 \]

Thus:

\[ 3 f'(1) = -15 \]

Dividing both sides by 3:

\[ f'(1) = -5 \]

### Final Answer

So, the value of \( f'(1) \) is:

\[ \boxed{-5} \]