Dig the Range

Geometry Level 5

f(x)=(cos1x2)2+πsin1x2(sin1x2)2+π212(x2+6x+8) \large \displaystyle f(x)=\left(\cos^{-1}\dfrac{x}{2}\right)^2+\pi \sin^{-1}\dfrac{x}{2}-\left(\sin^{-1}\dfrac{x}{2}\right)^2+\dfrac{\pi^2}{12}(x^2+6 x+8)

If the range of f(x)f(x) above is [aπ2,bπ2][a\pi^2,b \pi^2], what is the value of 2(a+b)2(a+b)?

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