The abscissae of the points,where the tangent to the curve $y = x^3 - 3x^2 - 9x + 5$ is parallel to the $x$-axis,are

If $f(x)=x^2+ax+b$ has a minima at $x=3$ whose value is $5$,then the values of $a$ and $b$ are respectively.

Find the absolute maximum and minimum values of the function $f$ given by $f(x) = \cos^{2} x + \sin x$,where $x \in [0, \pi]$.

Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi-vertical angle $\alpha$ is one-third that of the cone and the greatest volume of the cylinder is $\frac{4}{27} \pi h^{3} \tan^{2} \alpha$.

The two parts of $100$ for which the sum of double of the first part and the square of the second part is minimum,are

$A$ rectangle has one side on the positive $y-$ axis and one side on the positive $x-$ axis. The upper right hand vertex lies on the curve $y = \frac{\ln x}{x^2}$. The maximum area of the rectangle is