Differentiability with Floor Function

Calculus Level 4

f(x)={sin(x2π)x23x18+ax3+bfor 0x12cos(πx)+tan1xfor 1x2f(x) = \begin{cases} \dfrac{\sin (\lfloor x^2 \rfloor \pi)}{x^2-3x-18} +ax^3+b & \text{for } 0 \le x \le 1 \\ 2 \cos (\pi x)+ \tan^{-1} x & \text{for } 1 \le x \le 2 \end{cases}

If f(x)f(x) is differentiable in [0,2][0,2], find the value of π4ba\left| \dfrac{\pi}{4} - b - a \right| .

Note:

  • \lfloor \cdot \rfloor denotes the floor function
  • |\cdot | denotes the modulus function.
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