Consider the following for the next two (02) items that follow:

A differentiable function $f(x)$ has a local maximum at $x = 0$. Let $y = 2f(x) + ax - b$.

The function $y$ has a relative maxima at $x = 0$ for

Let $f(x)=|x-1|, g(x)=[x]$ and $h(x)=f(x) g(x)$ where [.] is greatest integer function.

What is $\int\limits_{-1}^{0} h(x) dx$ equal to?

Let $f(x)=|x-1|, g(x)=[x]$ and $h(x)=f(x) g(x)$ where [.] is greatest integer function.

What is $\int_{0}^{2} h(x) dx$ equal to?

Let $\int \frac{d x}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{\frac{3}{2}}+$ $$ \beta(x-1)^{\frac{3}{2}}+c $$

What is the value of $\alpha$?