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In subpoint (3), we are given the following problem:

\[
f''(x) = 2x - 6, \quad f(0) = 1, \quad f'(1) = 0.
\]

We need to determine the function \( f(x) \) that satisfies these conditions.

### Step-by-Step Solution

1. **Integrate \( f''(x) = 2x - 6 \) to find \( f'(x) \):**
   \[
   f'(x) = \int (2x - 6) \, dx = x^2 - 6x + C_1,
   \]
   where \( C_1 \) is a constant.

2. **Use \( f'(1) = 0 \) to find \( C_1 \):**
   Substitute \( x = 1 \) and \( f'(1) = 0 \):
   \[
   0 = (1)^2 - 6 \cdot 1 + C_1,
   \]
   \[
   0 = 1 - 6 + C_1,
   \]
   \[
   C_1 = 5.
   \]
   Thus, \( f'(x) = x^2 - 6x + 5 \).

3. **Integrate \( f'(x) = x^2 - 6x + 5 \) to find \( f(x) \):**
   \[
   f(x) = \int (x^2 - 6x + 5) \, dx = \frac{x^3}{3} - 3x^2 + 5x + C_2,
   \]
   where \( C_2 \) is a constant.

4. **Use \( f(0) = 1 \) to find \( C_2 \):**
   Substitute \( x = 0 \) and \( f(0) = 1 \):
   \[
   1 = \frac{0^3}{3} - 3 \cdot 0^2 + 5 \cdot 0 + C_2,
   \]
   \[
   C_2 = 1.
   \]
   Thus, \( f(x) = \frac{x^3}{3} - 3x^2 + 5x + 1 \).

### Final Answer
The function \( f(x) \) that satisfies all given conditions is:
\[
f(x) = \frac{x^3}{3} - 3x^2 + 5x + 1.
\]

Would you like a detailed explanation of any step?

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Here are some related questions to explore:

1. How do you verify that a solution satisfies initial conditions?
2. What techniques can be used for integrating polynomial functions?
3. How do higher-order derivatives influence the behavior of a function?
4. What role do integration constants play in solving differential equations?
5. How can you generalize this process for different forms of \( f''(x) \)?

**Tip:** When solving for a function with given derivative properties, work backwards from the highest derivative given, applying constants as you integrate each step.

Subpunctul 3: Determine the twice differentiable function f such that f''(x) = 2x - 6, f(0) = 1, and f'(1) = 0.

Learn how to determine a twice differentiable function f given that f''(x) = 2x - 6, with initial conditions f(0) = 1 and f'(1) = 0. This problem explores integration techniques, initial condition application, and the Fundamental Theorem of Calculus.

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