To determine an appropriate viewing window for the graph of the function $y = \frac{1}{3} x^3 - \frac{5}{2} x^2 + 6x - 4$, we need to consider both the function's behavior and the provided information in the question.

Key Information:

• The function has an intersection at $(1, 0)$.
• It rises from $(-1, -10)$ to $(2, 0.7)$, then falls to $(3, 0.5)$, and finally rises to $(5.5, 10)$.
• The window you are looking for should contain this range and also give enough space to observe the critical points and behavior of the function.

Step-by-Step Evaluation:

1. X-Values: Based on the provided description, the function is rising and falling between $x = -1$ to $x = 5.5$, so we want to make sure that the viewing window in the $x$-direction (horizontal axis) covers this range. An appropriate range for the $x$-axis could be from $x = 0$ to $x = 5$.

2. Y-Values: From the points described, the $y$-values vary from approximately $-10$ at $x = -1$ to around $10$ at $x = 5.5$. Therefore, we want the $y$-axis to range from $y = -20$ to $y = 20$ to comfortably include all critical values and behavior.

Conclusion:

The most appropriate window is Option B, which covers the range for $x$ from $0$ to $5$ and the range for $y$ from $-20$ to $20$.

Answer:

B. $[0, 5]$ by $[-20, 20]$

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Would you like more details on how to identify extreme points of the graph?

Related Questions:

1. How do you find the critical points of a cubic function?
2. What does the first derivative test tell us about a function's behavior?
3. How do we determine the global maximum and minimum of a function?
4. Can the second derivative be used to identify concavity and inflection points?
5. How do the inflection points of this cubic function relate to the graph's shape?

Tip:

When choosing a viewing window for graphing, ensure that the window captures the important features like the function’s turning points, intersections, and extreme values without cutting off any key parts of the graph.