If $x+y=k, x>0, y>0$,then $x^2+y^2$ is minimum,if

Let $P(x)$ be a polynomial of degree $3$ having an extreme value at $x=1$. If $\lim _{x \rightarrow 0}\left(\frac{P(x)+4}{x^2}+2\right)=6$, then $\left(\frac{d P}{d x}\right)_{x=\frac{1}{2}}=$

The greatest value of the function $f(x) = -1 + \frac{2}{2^{x^2} + 1}$ is:

Find the maximum and minimum values of the function given by $h(x) = x + 1, x \in (-1, 1)$.

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,

For the function $f(x)=x^3-6x^2-12x-3$,$x=2$ is a