If $$x=a(\theta+\sin \theta)$$ and $$y=a(1-\cos \theta)$$ then $$\left(\frac{d^2 y}{d x^2}\right)_{at~ \theta=\pi / 2}=$$

The equation of tangent to the curve $$y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)$$ at $$x=\frac{\pi}{4}$$, is

Function $$f(x)=e^{-1 / x}$$ is strictly increasing for all $$x$$ where

If $$x=-2$$ and $$x=4$$ are the extreme points of $$y=x^3-\alpha x^2-\beta x+5$$, then