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We are given the function \( f(x) = x^3 + kx^2 + 3x + 1 \), where \( k \) is a constant, and we are asked to determine the values of \( k \) so that \( f(x) \) is an increasing function ("voksende funktion" in Danish).

### Step 1: Find the derivative of the function
To analyze whether the function is increasing, we need to look at its derivative, \( f'(x) \). A function is increasing when its derivative is positive, i.e., \( f'(x) > 0 \).

The derivative of \( f(x) \) is:

\[
f'(x) = 3x^2 + 2kx + 3
\]

### Step 2: Analyze when the derivative is positive
We need to find when the derivative \( f'(x) \) is positive for all \( x \). Specifically, we want to find the conditions on \( k \) such that \( 3x^2 + 2kx + 3 > 0 \) for all \( x \).

The discriminant \( \Delta \) of the quadratic \( 3x^2 + 2kx + 3 = 0 \) is:

\[
\Delta = (2k)^2 - 4(3)(3) = 4k^2 - 36
\]

For the quadratic to have no real roots (so that it does not cross the x-axis and remains positive), we require \( \Delta < 0 \):

\[
4k^2 - 36 < 0
\]

Simplifying this:

\[
4k^2 < 36
\]

\[
k^2 < 9
\]

\[
-3 < k < 3
\]

### Conclusion:
For the function \( f(x) = x^3 + kx^2 + 3x + 1 \) to be increasing for all \( x \), the constant \( k \) must satisfy \( -3 < k < 3 \).

Would you like further clarification or details on this? Here are five related questions you might be interested in:

1. How do you find the derivative of a cubic function?
2. What is the significance of the discriminant in determining the behavior of a quadratic equation?
3. How can we visually interpret the condition \( f'(x) > 0 \) on a graph?
4. How does the value of \( k \) affect the shape of the cubic function?
5. What is the difference between an increasing and strictly increasing function?

**Tip:** When analyzing the behavior of polynomials, the sign of the derivative helps identify intervals where the function is increasing or decreasing.

Bestem de værdier af k, så så f er en voksende funktion

Solve the problem of finding the range of values for k that ensures the function f(x) = x^3 + kx^2 + 3x + 1 is increasing. This involves calculating the derivative, using the discriminant, and solving inequalities. Suitable for high school calculus students.

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