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The problem on the screen is related to rotating the graph of a function \( f(x) = 8x + e^{\frac{6x - 9x^2}{2}} \) counterclockwise or clockwise through an angle \( \theta \) and determining the largest value of \( \theta \) for which the rotated graph is still the graph of a function.

### Steps to Solve:

1. **Maximal and Minimal Slopes**: 
   We need to find the **maximal slope** and **minimal slope** of the function \( f(x) \). These correspond to the steepest tangent lines on the graph. 
   
   The slope of the tangent line to \( f(x) \) is given by its derivative, \( f'(x) \). We will compute the derivative of \( f(x) \), and then find its maximum and minimum values.

   The function is \( f(x) = 8x + e^{\frac{6x - 9x^2}{2}} \).

   To find \( f'(x) \), apply the chain rule:

   \[
   f'(x) = \frac{d}{dx}\left(8x + e^{\frac{6x - 9x^2}{2}}\right) = 8 + e^{\frac{6x - 9x^2}{2}} \cdot \frac{d}{dx} \left(\frac{6x - 9x^2}{2}\right)
   \]

   Simplifying the derivative of the exponent:

   \[
   \frac{d}{dx} \left(\frac{6x - 9x^2}{2}\right) = 3 - 9x
   \]

   Therefore:

   \[
   f'(x) = 8 + e^{\frac{6x - 9x^2}{2}} \cdot (3 - 9x)
   \]

2. **Determine the Maximal and Minimal Values of \( f'(x) \)**:
   We now need to analyze the behavior of this derivative function to find its maximal and minimal values. This can be done by finding critical points (where \( f''(x) = 0 \)) and examining the limits of \( f'(x) \) as \( x \to \infty \) or \( x \to -\infty \), but in general, computational tools can help graph this function and find the extrema.

3. **Finding the Rotation Angle \( \theta \)**:
   The largest rotation angle \( \theta \) (for counterclockwise rotation) depends on the maximal slope. If the maximal slope is \( m_{\text{max}} \), then the tangent of the rotation angle is given by:

   \[
   \theta_{\text{max}} = \tan^{-1}(m_{\text{max}})
   \]

   Similarly, for clockwise rotation, the largest angle corresponds to the minimal slope \( m_{\text{min}} \), and the clockwise rotation angle will be:

   \[
   \theta_{\text{min}} = \tan^{-1}(m_{\text{min}})
   \]

4. **Interpretation of Angles**:
   - The maximal counterclockwise angle corresponds to the maximum slope (the steepest positive slope).
   - The maximal clockwise angle corresponds to the minimum slope (the steepest negative slope), and the angle is expressed as a negative value to represent clockwise rotation.

### Conclusion:
You can proceed by calculating the maximal and minimal values of the derivative, and then find the corresponding angles \( \theta \) using the arctangent function.

Would you like help computing the maximal and minimal slopes or further steps with the derivative calculations? 

### Relative Questions:
1. How can the chain rule be applied to differentiate complex functions like \( f(x) \)?
2. What is the significance of critical points in determining the extrema of a function?
3. How do you interpret the derivative in terms of the graph's steepness?
4. What role does the arctangent function play in converting slopes to angles?
5. How does the Mean Value Theorem help analyze function behaviors like extrema?

**Tip**: When solving problems involving rotations of graphs, always consider how the slope changes at different points on the graph.

What is the largest value of θ for which the rotated graph of f(x) = 8x + e^{(6x - 9x^2)/2} is still the graph of a function? How does this change if the graph is rotated clockwise?

Explore how to find the maximal value of θ for which the graph of f(x) = 8x + e^{(6x - 9x^2)/2} can be rotated while still remaining a function. This problem involves differentiation, trigonometric relations, and graph rotation concepts suitable for advanced high school or college students.

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