The problem is asking to find the antiderivative $F(x)$ of the function $f(x) = 3x^2 - 4x$, and it provides the following conditions for the values of the antiderivative at specific points:

• $F(-1) = 0$
• $F(0) = 0$
• $F(3) = 0$

Step 1: Find the general antiderivative

To begin, let's compute the general antiderivative $F(x)$ of the given function $f(x) = 3x^2 - 4x$.

$F(x) = \int (3x^2 - 4x) \, dx$

For each term:

• The antiderivative of $3x^2$ is $x^3$,
• The antiderivative of $-4x$ is $-2x^2$.

Thus, the general form of the antiderivative is:

$F(x) = x^3 - 2x^2 + C$

where $C$ is the constant of integration.

Step 2: Use the conditions to find $C$

Now, we use the given conditions to determine the value of $C$. The conditions are:

• $F(-1) = 0$
• $F(0) = 0$
• $F(3) = 0$

Let's apply each of these conditions.

1. $F(-1) = (-1)^3 - 2(-1)^2 + C = -1 - 2 + C = C - 3 = 0$, so $C = 3$.
2. $F(0) = 0^3 - 2(0)^2 + 3 = 3$, but the condition states $F(0) = 0$, which contradicts this.
3. $F(3) = 3^3 - 2(3)^2 + 3 = 27 - 18 + 3 = 12$, but this also contradicts $F(3) = 0$.

It appears there may be an inconsistency in the conditions as given, but based on the standard process, we would conclude that the correct constant $C = 3$, though it does not satisfy all conditions simultaneously.

Would you like further clarification or another approach to check the issue with the conditions?

Relative Questions:

1. What are the general rules for finding antiderivatives of polynomials?
2. How would we solve if the conditions were adjusted for consistency?
3. What is the importance of boundary conditions in determining specific solutions for integrals?
4. Can a function have more than one antiderivative that fits given conditions?
5. How would you approach integrating functions involving trigonometric terms?

Tip:

When solving for an antiderivative with conditions, ensure that all conditions are consistent. If they lead to contradictions, reconsider the initial assumptions or conditions provided.