If $$x=f(\theta)$$ and $$y=g(\theta)$$, then $$\frac{d^2 y}{d x^2}=$$

$$y=x^3-a x^2+48 x+7$$ is an increasing function for all real values of $$x$$, then $$a$$ lies in the interval

If $$x \neq 0$$ and $$f(x)$$ satisfies $$8 f(x)+6 f(1 / x) =x+5$$, then $$\frac{d}{d x}\left(x^2 f(x)\right)$$ at $$x=1$$ is

If $$f(x)=\cot ^{-1}\left(\frac{x^x+x^{-x}}{2}\right)$$, then $$f^{\prime}(1)=$$