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Q.
If the range of the function $f(x)=\tan ^{-1}\left(3 x^2+b x+c\right)$ is $\left[0, \frac{\pi}{2}\right)$ then
Inverse Trigonometric Functions
Solution:
Range of $f(x)=\tan ^{-1}\left(3 x^2+b x+c\right)$ is $\left[0, \frac{\pi}{2}\right)$ if and only if range of $g(x)=3 x^2+b x+c$ is $[0, \infty)$.
This is possible only when discriminant of the equation $3 x ^2+ bx + c =0$ is equal to zero. i.e. $b^2=12c$