Let $f(x)=x^2+a x+b$,where $a, b \in R$. If $f(x)=0$ has all its roots imaginary,then the roots of $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=0$ are

Let $f: ( -\infty, \infty ) \to ( -\infty, \infty )$ be defined by $f(x) = x^3 + 1$.
Statement $1$: The function $f$ has a local extremum at $x = 0$.
Statement $2$: The function $f$ is continuous and differentiable on $( -\infty, \infty )$ and $f'(0) = 0$.

Observe the following statements:
$I. f(x) = a x^{41} + b x^{-40} \Rightarrow \frac{f^{\prime \prime}(x)}{f(x)} = 1640 x^{-2}$
$II. \frac{d}{d x} \tan ^{-1}\left(\frac{2 x}{1-x^2}\right) = \frac{1}{1+x^2}$
Which of the following is correct?

Let $f$ and $g$ be twice differentiable functions on $R$ such that
$f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x$
$f^{\prime}(1)=4, g^{\prime}(1)=3$
$f(2)=12, g(2)=4$
Then which of the following is $NOT$ true?

$\mathop {\lim }\limits_{x \to 0} \frac{d}{{dx}}\left( {\frac{{{e^{{e^{{x^2}}}}} - e}}{x}} \right)$ is

Let $f:R \to R$ be a continuously differentiable function such that $f(2) = 6$ and $f'(2) = \frac{1}{48}$. If $\int_6^{f(x)} 4t^3 \,dt = (x - 2)g(x)$,then $\lim_{x \to 2} g(x)$ is equal to