A solution curve of the differential equation

$$\left( {{x^2} + xy + 4x + 2y + 4} \right){{dy} \over {dx}} - {y^2} = 0,$$ $$x>0,$$ passes through the

point $$(1,3)$$. Then the solution curve

The least value of a $$ \in R$$ for which $$4a{x^2} + {1 \over x} \ge 1,$$, for all $$x>0$$. is

In a triangle $$\Delta $$$$XYZ$$, let $$x, y, z$$ be the lengths of sides opposite to the angles $$X, Y, Z$$ respectively, and $$2s = x + y + z$$.
If $${{s - x} \over 4} = {{s - y} \over 3} = {{s - z} \over 2}$$ and area of incircle of the triangle $$XYZ$$ is $${{8\pi } \over 3}$$, then

Let $$f:\mathbb{R} \to \mathbb{R},\,g:\mathbb{R} \to \mathbb{R}$$ and $$h:\mathbb{R} \to \mathbb{R}$$ be differentiable functions such that $$f\left( x \right)= {x^3} + 3x + 2,$$ $$g\left( {f\left( x \right)} \right) = x$$ and $$h\left( {g\left( {g\left( x \right)} \right)} \right) = x$$ for all $$x \in R$$. Then