The function given is:

$g(t) = -\frac{2}{9}t^4 + 3t$

Left-hand and right-hand behavior:

1. As $t \to -\infty$ (left-hand behavior):

   • The highest degree term dominates the behavior of the function as $t \to -\infty$. The term $-\frac{2}{9}t^4$ grows very large negatively, since $t^4$ will grow large and positive, but the negative sign in front makes the whole term negative.
   • The linear term $3t$ contributes a smaller effect because $t^4$ dominates. Therefore, as $t \to -\infty$, $g(t) \to -\infty$.

2. As $t \to \infty$ (right-hand behavior):

   • Similarly, for $t \to \infty$, the term $-\frac{2}{9}t^4$ dominates. Since $t^4$ becomes very large and the negative coefficient makes it negative, the function will tend to $-\infty$ as well.
   • The linear term $3t$ again has a smaller influence in comparison to the dominant $t^4$-term. Thus, as $t \to \infty$, $g(t) \to -\infty$.

Critical points:

To better understand the behavior and find any critical points, we differentiate $g(t)$ with respect to $t$:

$g'(t) = -\frac{8}{9}t^3 + 3$

Setting $g'(t) = 0$ to find critical points:

$-\frac{8}{9}t^3 + 3 = 0$ $t^3 = \frac{27}{8}$ $t = \sqrt[3]{\frac{27}{8}} = \frac{3}{2}$

So, $t = \frac{3}{2}$ is a critical point. We can analyze the second derivative to classify the critical point as a local maximum or minimum.

$g''(t) = -\frac{24}{9}t^2 = -\frac{8}{3}t^2$

At $t = \frac{3}{2}$:

$g''\left( \frac{3}{2} \right) = -\frac{8}{3} \times \left( \frac{3}{2} \right)^2 = -6$

Since $g''(t) < 0$, the function has a local maximum at $t = \frac{3}{2}$.

Sketch of the graph:

• The function has a local maximum at $t = \frac{3}{2}$, and the value of $g(t)$ at this point can be found by substituting $t = \frac{3}{2}$ into the original function.
• The graph falls towards $-\infty$ on both sides (left and right behavior), with the local maximum in between.

Let me know if you'd like me to generate the graph.

Do you want any details or clarifications on this?

Relative questions:

1. What is the significance of critical points in understanding a function's graph?
2. How does the degree of a polynomial affect its end behavior?
3. What is the importance of the second derivative in determining concavity?
4. Can a function have more than one local maximum or minimum?
5. How do inflection points relate to the curvature of a graph?

Tip: When analyzing polynomials, the highest degree term typically dictates the end behavior of the graph.