The value of $$\int\limits_1^a {\left[ x \right]} f'\left( x \right)dx,a > 1$$ where $${\left[ x \right]}$$ denotes the greatest integer not exceeding $$x$$ is

The set of points where $$f\left( x \right) = {x \over {1 + \left| x \right|}}$$ is differentiable is

If $${x^m}.{y^n} = {\left( {x + y} \right)^{m + n}},$$ then $${{{dy} \over {dx}}}$$ is

The function $$f\left( x \right) = {x \over 2} + {2 \over x}$$ has a local minimum at