Rotating a Graph?

Calculus Level 5

The above shows twos functions on a same Cartesian Plane: \( f(x) \) and \(f^\theta (x) \) colored blue and red respectively.


Let \(f(x)=x^3-6x^2+6x+1\)

Define \(f^{\theta}(x)\), where \(\theta \geq 0\), as the new graph obtained by rotating the graph of \(f(x)\) \(\theta\) degrees about the origin (\(0,0\)) in the clockwise direction.

Find the maximum value of \(\theta_M\) such that for all values of \(0<\theta<\theta_M\), \(f^{\theta}(x)\) has every value of \(x\) paired with only one \(y\) (more simply, no two points can have the same \(x\)-coordinate).

Input your answer as \(\lfloor 100\cdot\theta_m \rfloor \). Where \(\lfloor x \rfloor\) is the floor function which means the greatest integer less than or equal to \(x\).

This is part of the set Trevor's Ten

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