Let `f(x)` have a point of inflection at `x=1` and let `f^(")(x)=x.` If `f^(')(1)=0,` then `f(x)` is equal to

Options

A. `x^(4)-6x^(2)+8x`
B. `x^(4)-6x^(2)-x`
C. `1/24 (x^(4)-6x^(2)+8x)`
D. `1/12(x^(4)-6x^(2)+8x)`

Explanation

To solve the problem, we need to find the function \( f(x) \) given that \( f''(x) = x \), \( f'(1) = 0 \), and there is a point of inflection at \( x = 1 \).

### Step-by-Step Solution:

1. **Integrate \( f''(x) \)**:
We start with the second derivative of the function:
\[ f''(x) = x \]
To find the first derivative \( f'(x) \), we integrate \( f''(x) \):
\[ f'(x) = \int x \, dx = \frac{x^2}{2} + C_1 \]
where \( C_1 \) is a constant.

**Hint**: Remember that integrating \( x \) gives \( \frac{x^2}{2} \).

2. **Use the condition \( f'(1) = 0 \)**:
We know that \( f'(1) = 0 \):
\[ f'(1) = \frac{1^2}{2} + C_1 = 0 \]
This simplifies to:
\[ \frac{1}{2} + C_1 = 0 \implies C_1 = -\frac{1}{2} \]
Therefore, we have:
\[ f'(x) = \frac{x^2}{2} - \frac{1}{2} \]

**Hint**: Substitute \( x = 1 \) into \( f'(x) \) to find \( C_1 \).

3. **Integrate \( f'(x) \)**:
Now we integrate \( f'(x) \) to find \( f(x) \):
\[ f(x) = \int \left( \frac{x^2}{2} - \frac{1}{2} \right) dx = \frac{x^3}{6} - \frac{x}{2} + C_2 \]
where \( C_2 \) is another constant.

**Hint**: When integrating, remember to add a constant \( C_2 \).

4. **Determine the constant \( C_2 \)**:
Since we know that there is a point of inflection at \( x = 1 \), we can use the fact that \( f''(1) = 1 \) (since \( f''(x) = x \)):
\[ f''(1) = 1 \]
This condition is already satisfied by our \( f''(x) \), so we can choose \( C_2 \) freely. For simplicity, let's set \( C_2 = 0 \):
\[ f(x) = \frac{x^3}{6} - \frac{x}{2} \]

**Hint**: The point of inflection condition confirms the behavior of the second derivative.

5. **Final form of \( f(x) \)**:
Thus, the function \( f(x) \) is:
\[ f(x) = \frac{x^3}{6} - \frac{x}{2} + C_2 \]
If we set \( C_2 = 0 \), we have:
\[ f(x) = \frac{x^3}{6} - \frac{x}{2} \]

### Conclusion:
The final answer for \( f(x) \) is:
\[ f(x) = \frac{x^3}{6} - \frac{x}{2} \]