If $$y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ is

The population $$\mathrm{P}=\mathrm{P}(\mathrm{t})$$ at time $$\mathrm{t}$$ of certain species follows the differential equation $$\frac{d P}{d t}=0.5 P-450$$. If $$P(0)=850$$, then the time at which population becomes zero is

The vertices of the feasible region for the constraints $$x+y \leq 4, x \leq 2, y \leq 1, x+y \geq 1, x, y \geq 0$$ are

The value of $$\tan \frac{\pi}{8}$$ is