Let the radius and height of a right circular cylinder be related as $r^2 + h = 6$. If the volume of the cylinder is maximum,then the value of $\frac{r}{h}$ is:

Let a function $f:[0,5] \rightarrow R$ be continuous. $f(1)=3$ and $F$ be defined as $F(x)=\int_{1}^{x} t^{2} g(t) dt$,where $g(t)=\int_{1}^{t} f(u) du$. Then for the function $F$,the point $x=1$ is

If $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$,then the value of $x$ for which $f(x)$ is minimum is:

At $x=0$,$f(x)=\cos x-1+\frac{x^2}{2}-\frac{x^3}{3}$

Let $P(x)$ be a real polynomial of degree $3$ which vanishes at $x = -3$. Let $P(x)$ have local minima at $x = 1$,local maxima at $x = -1$,and $\int_{-1}^{1} P(x) dx = 18$. Then the sum of all the coefficients of the polynomial $P(x)$ is equal to ....... .

$A$ cone of maximum volume is inscribed in a given sphere. Then the ratio of the height of the cone to the diameter of the sphere is: