If the range of the function \( f(x)=\tan ^{-1}\left(3 x^{2}+b x+c\right) \) is \( \left[0, \fra...

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If the range of the function \( f(x)=\tan ^{-1}\left(3 x^{2}+b x+c\right) \) is \( \left[0, \frac{\pi}{2}\right) \); (domain is \( R \) ), then:
(a) \( b^{2}=3 c \)
(b) \( b^{2}=4 c \)
(c) \( b^{2}=12 c \)
(d) \( b^{2}=8 c \)
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