Here’s a function problem for you:

Problem:

Let $f(x) = 2x^3 - 3x^2 + 4x - 5$.

1. Find the value of $f(2)$.
2. Determine the first derivative, $f'(x)$.
3. Find the critical points of $f(x)$.
4. Determine the intervals where $f(x)$ is increasing or decreasing.
5. Find the local minima and maxima of $f(x)$.
6. Compute the second derivative, $f''(x)$.
7. Determine the concavity of the function and find any inflection points.
8. Calculate the y-intercept and find the roots of $f(x)$ (if any).

Would you like details on how to solve any of these parts, or do you have any questions?

Here are 8 related questions for further exploration:

1. How do you find the critical points of a function?
2. What is the significance of the first derivative in determining the behavior of a function?
3. How do you determine where a function is increasing or decreasing?
4. What is the second derivative test, and how is it used to classify critical points?
5. What are inflection points, and how do you find them?
6. How do you solve a cubic equation like $2x^3 - 3x^2 + 4x - 5 = 0$?
7. What is the geometric interpretation of the first and second derivatives of a function?
8. How do you find the exact roots of a cubic polynomial?

Tip: When solving for critical points, always remember to check both the first and second derivatives to classify the nature of each critical point.