f(x)=-3x^3+10x^2+9x-4

1. bestem, ved aflæsning af grafen, monotoniforhold, ekstrema og krumningsforhold for f

To analyze the function \( f(x) = -3x^3 + 10x^2 + 9x - 4 \) and determine the monotonicity, extrema, and concavity, let's proceed with each step:

### 1. Monotonicity
Monotonicity refers to where the function is increasing or decreasing. To analyze this, we need the first derivative, \( f'(x) \), and find its sign over different intervals.

#### First Derivative:
\[ f'(x) = -9x^2 + 20x + 9 \]

To find where the function changes from increasing to decreasing (or vice versa), set \( f'(x) = 0 \):
\[
-9x^2 + 20x + 9 = 0
\]
This is a quadratic equation in \( x \), and solving it gives us the critical points.

#### Solving \( f'(x) = 0 \):
Use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = -9 \), \( b = 20 \), and \( c = 9 \). Calculating this, we get:
\[
x = \frac{-20 \pm \sqrt{400 + 324}}{-18}
\]
\[
x \approx 3 \text{ and } x \approx -\frac{1}{3}
\]

These points divide the x-axis into intervals. By testing the sign of \( f'(x) \) in each interval, we can determine if \( f(x) \) is increasing or decreasing in each interval.

### 2. Extrema
The critical points found above (where \( f'(x) = 0 \)) are potential extrema. We check the values of \( f(x) \) at these points to determine whether they are local maxima or minima.

- If \( f'(x) \) changes from positive to negative at a critical point, it's a local maximum.
- If \( f'(x) \) changes from negative to positive, it's a local minimum.

By substituting values around \( x \approx 3 \) and \( x \approx -\frac{1}{3} \), we determine the nature of each extremum.

### 3. Concavity and Inflection Points
To analyze concavity, we use the second derivative \( f''(x) \).

#### Second Derivative:
\[ f''(x) = -18x + 20 \]

Setting \( f''(x) = 0 \) gives the inflection point(s), where the concavity changes.

\[
-18x + 20 = 0
\]
\[
x = \frac{20}{18} = \frac{10}{9} \approx 1.11
\]

At \( x \approx 1.11 \), the concavity of \( f(x) \) changes:
- If \( f''(x) > 0 \) to the left of \( x = 1.11 \), the function is concave up.
- If \( f''(x) < 0 \) to the right of \( x = 1.11 \), the function is concave down.

### Summary
- **Monotonicity**: Determine intervals of increase/decrease using \( f'(x) \).
- **Extrema**: Evaluate \( f(x) \) at critical points to find local maxima or minima.
- **Concavity**: Use \( f''(x) \) to determine intervals of concave up/down and identify inflection points.

Would you like a more detailed analysis or a graph of these findings?

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### Further Exploration Questions
1. How would the graph change if the coefficient of \( x^3 \) were positive?
2. How does the second derivative test help determine concavity?
3. Could we approximate extrema and inflection points without calculus? If so, how?
4. What real-world scenarios might this cubic function model?
5. How would the function behave as \( x \to \infty \) or \( x \to -\infty \)?

### Tip:
Analyzing the signs of the first and second derivatives on intervals can quickly give insights into the function's behavior.

bestem, ved aflæsning af grafen, monotoniforhold, ekstrema og krumningsforhold for f(x) = -3x^3 + 10x^2 + 9x - 4

Explore the detailed analysis of the function f(x) = -3x^3 + 10x^2 + 9x - 4. Learn how to determine the monotonicity, extrema, and concavity through the first and second derivatives, including finding critical points and inflection points. Suitable for high school and college-level students.

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