To analyze the function \( f(x) = 2x^3 - 5x^2 - 19x + 42 \), we can discuss its key features such as critical points, behavior, and roots:

1. **Critical Points**: These occur where the first derivative \( f'(x) \) is equal to zero.
   \[
   f'(x) = \frac{d}{dx}(2x^3 - 5x^2 - 19x + 42) = 6x^2 - 10x - 19
   \]
   Now, solve \( 6x^2 - 10x - 19 = 0 \) to find the critical points.

2. **Inflection Points**: These occur where the second derivative \( f''(x) \) is equal to zero.
   \[
   f''(x) = \frac{d}{dx}(6x^2 - 10x - 19) = 12x - 10
   \]
   Solve \( 12x - 10 = 0 \) to find the inflection point.

3. **Roots**: The roots of the function are the values of \( x \) for which \( f(x) = 0 \). To find these, solve the cubic equation:
   \[
   2x^3 - 5x^2 - 19x + 42 = 0
   \]

4. **End Behavior**: Since the leading term is \( 2x^3 \), the function behaves like a cubic function at extreme values of \( x \). As \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).

Would you like me to calculate any specific features, such as the roots or critical points?

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Here are 5 related questions:
1. How do we find the maximum or minimum values of the function?
2. What is the behavior of the function at the critical points?
3. Can you graph this cubic function based on its roots and critical points?
4. How do you solve cubic equations like \( 2x^3 - 5x^2 - 19x + 42 = 0 \)?
5. How do inflection points affect the graph of a function?

**Tip**: When analyzing polynomial functions, always consider derivatives to find critical and inflection points, as they reveal much about the graph's shape.

Learn how to analyze the cubic function f(x) = 2x^3 - 5x^2 - 19x + 42, including finding critical points, inflection points, and the behavior of the function. This problem covers calculus concepts such as derivatives and solving cubic equations. Suitable for high school students in Grades 10-12.

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