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.