If `f'(x)=8x^(3)+3x^(2)-10x-k " and " f(0)=-3,f(-1)=0," then " f(x)=` is

Options

A. `2x^(4)+x^(3)+5x^(2)+7x-3`
B. `2x^(4)+x^(3)+5x^(2)-7x+3`
C. `2x^(4)+x^(3)-5x^(2)+7x-3`
D. `2x^(4)+x^(3)-5x^(2)-7x-3`

Answer

To find the function \( f(x) \) given that \( f'(x) = 8x^3 + 3x^2 - 10x - k \) and the conditions \( f(0) = -3 \) and \( f(-1) = 0 \), we will follow these steps:

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

We start by integrating \( f'(x) \) to find \( f(x) \):
\[ f(x) = \int (8x^3 + 3x^2 - 10x - k) \, dx \]

### Step 2: Perform the Integration

Integrating each term separately:
\[ f(x) = \int 8x^3 \, dx + \int 3x^2 \, dx - \int 10x \, dx - \int k \, dx \]
\[ = 2x^4 + x^3 - 5x^2 - kx + C \]

### Step 3: Use the Condition \( f(0) = -3 \)

Now we use the condition \( f(0) = -3 \):
\[ f(0) = 2(0)^4 + (0)^3 - 5(0)^2 - k(0) + C = -3 \]
This simplifies to:
\[ C = -3 \]

### Step 4: Substitute \( C \) into \( f(x) \)

Now we substitute \( C \) back into the equation for \( f(x) \):
\[ f(x) = 2x^4 + x^3 - 5x^2 - kx - 3 \]

### Step 5: Use the Condition \( f(-1) = 0 \)

Next, we use the condition \( f(-1) = 0 \):
\[ f(-1) = 2(-1)^4 + (-1)^3 - 5(-1)^2 - k(-1) - 3 = 0 \]
Calculating each term:
\[ = 2(1) - 1 - 5 + k - 3 = 0 \]
This simplifies to:
\[ 2 - 1 - 5 - 3 + k = 0 \]
\[ k - 7 = 0 \implies k = 7 \]

### Step 6: Substitute \( k \) back into \( f(x) \)

Now we substitute \( k = 7 \) back into the equation for \( f(x) \):
\[ f(x) = 2x^4 + x^3 - 5x^2 - 7x - 3 \]

### Final Answer

Thus, the function \( f(x) \) is:
\[ \boxed{2x^4 + x^3 - 5x^2 - 7x - 3} \]