The feasible region represented by the given constraints $2 x+3 y \geq 12,-x+y \leq 3, x \leq 4, y \geq 3$ is denoted by

For $\mathrm{n} \in \mathbb{N}$ if $y=\mathrm{a} x^{\mathrm{n}+1}+\mathrm{b} x^{-\mathrm{n}}$, then $x^2 \frac{\mathrm{~d}^2 y}{\mathrm{~d} x^2}=$

$$\begin{aligned} & \mathrm{f}(x)=(\cos x+\mathrm{i} \sin x) \cdot(\cos 3 x+\mathrm{i} \sin 3 x) \cdots {[\cos (2 \mathrm{n}-1) x+\mathrm{i} \sin (2 \mathrm{n}-1) x] \mathrm{n} \in \mathbb{N}} \end{aligned}$$

Then $\mathrm{f}^{\prime \prime}(x)=$ _______ , (Where $\mathrm{i}=\sqrt{-1}$ )

A population $p(t)$ of 1000 bacteria introduced into a nutrient medium grows according to the relation $\mathrm{p}(\mathrm{t})=1000+\frac{1000 \mathrm{t}}{100+\mathrm{t}^2}$. The maximum size of this bacterial population is