A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.

Let $$f\left( x \right) = \left\{ {\matrix{ {x{e^{ax}},\,\,\,\,\,\,\,x \le 0} \cr {x + a{x^2} - {x^3},\,x > 0} \cr } } \right.$$

Where a is a positive constant. Find the interval in which $$f'(x)$$ is increasing.

Determine the points of maxima and minima of the function
$$f\left( x \right) = {1 \over 8}\ell n\,x - bx + {x^2},x > 0,$$ where $$b \ge 0$$ is a constant.

Let $$(h, k)$$ be a fixed point, where $$h > 0,k > 0.$$. A straight line passing through this point cuts the possitive direction of the coordinate axes at the points $$P$$ and $$Q$$. Find the minimum area of the triangle $$OPQ$$, $$O$$ being the origin.