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The given problem asks us to find the antiderivative (or "stamfunktion" in Danish) of the function \( f(x) = 2x^3 - 6x \), and it specifies that the graph passes through the point \( P(1,10) \).

### Step 1: Find the general antiderivative of \( f(x) \)
To find the antiderivative, we integrate the given function:

\[
F(x) = \int (2x^3 - 6x) \, dx
\]

For each term:

1. \(\int 2x^3 \, dx = \frac{2}{4}x^4 = \frac{1}{2}x^4\)
2. \(\int -6x \, dx = -6 \cdot \frac{x^2}{2} = -3x^2\)

Thus, the general form of the antiderivative is:

\[
F(x) = \frac{1}{2}x^4 - 3x^2 + C
\]

where \( C \) is the constant of integration.

### Step 2: Use the point \( P(1,10) \) to find \( C \)
We know that \( F(1) = 10 \), so we substitute \( x = 1 \) and \( F(x) = 10 \) into the equation to find \( C \):

\[
10 = \frac{1}{2}(1)^4 - 3(1)^2 + C
\]

Simplifying:

\[
10 = \frac{1}{2} - 3 + C
\]

\[
10 = -\frac{5}{2} + C
\]

Now, solve for \( C \):

\[
C = 10 + \frac{5}{2} = \frac{20}{2} + \frac{5}{2} = \frac{25}{2}
\]

Thus, \( C = \frac{25}{2} \).

### Final answer:
The specific antiderivative is:

\[
F(x) = \frac{1}{2}x^4 - 3x^2 + \frac{25}{2}
\]

Let me know if you want more details on any steps, and here are some related questions:

1. How would you find the antiderivative of a more complex function?
2. What is the geometric meaning of the antiderivative in terms of the area under the curve?
3. How can you verify that \( F(x) \) is the correct antiderivative?
4. How would you apply similar methods if the function involved trigonometric terms?
5. What if the point \( P \) was different—how would that affect the result?

**Tip:** When integrating polynomials, always increase the power of the variable by 1 and divide by the new power!

Bestem den stamfunktion til f, hvis graf går gennem punktet P(1,10) for funktionen f(x) = 2x^3 - 6x.

Learn how to find the antiderivative of the function f(x) = 2x^3 - 6x, with detailed steps on integrating polynomials and determining the constant using a given point P(1,10). This calculus problem highlights key concepts in antiderivatives and polynomial functions, suitable for students in Grades 11-12.

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