Solving the provided function $G(x) = x^4 + 2x^3 - 3x^2 - 4x$:

1. Critical Points (First-Order Critical Numbers):

These are the points where the first derivative $G'(x) = 4x^3 + 6x^2 - 6x - 4$ equals zero:

• Critical points: $x = -2, -\frac{1}{2}, 1$.

2. Intervals of Monotonicity:

Monotonicity is determined by the sign of the first derivative $G'(x)$. The intervals where $G'(x)$ is positive indicate where the function is increasing, and where $G'(x)$ is negative indicate where the function is decreasing. These can be tested between the critical points found.

3. Local Extrema:

The local maxima or minima occur at the critical points. We check the sign of the first derivative before and after each critical point to determine if it is a local maximum or minimum.

4. Second-Order Critical Numbers:

Points where the second derivative $G''(x) = 12x^2 + 12x - 6$ equals zero:

• Second-order critical points (potential inflection points): $x \approx -0.5 + \frac{\sqrt{3}}{2}, -0.5 - \frac{\sqrt{3}}{2}$.

5. Intervals of Concavity:

The concavity of the function depends on the sign of the second derivative $G''(x)$. Positive $G''(x)$ indicates concave up, and negative $G''(x)$ indicates concave down.

6. Inflection Points:

These occur where the second derivative changes sign. From the second-order critical points above, these values indicate the points of inflection.

Graphing:

I'll generate the graph with the function, highlighting the critical points and inflection points.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like me to try explaining or calculating anything else about the function manually? Here are some follow-up questions to deepen your understanding:

1. How do we determine whether a critical point is a local maximum or minimum?
2. What are the steps to test the intervals of increasing and decreasing behavior for a function?
3. How can we use the second derivative test to confirm concavity?
4. Why do inflection points occur where the second derivative changes sign?
5. What is the significance of the points where both the first and second derivatives are zero?

Tip: Always check the sign of the derivatives to identify behavior changes like turning points and inflection points in a function.