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Given $f^{ \prime \prime }( x ) =4x-1$ and $f^{ \prime }( 0 ) =-1,$ at what point does $f(x)$ reach a minimum if $f(x)$ goes through the point $(0, 2)$?

$ (0, 2)$

$ \left(1, \cfrac{7}{6}\right)$

$ \left(1, -\cfrac{17}{6}\right)$

$ \left(-\cfrac{1}{2}, 0\right)$