The real valued function $f(x)=\frac{x^2}{2}-\log \left(x^2+x+1\right)$ is

If $x$ and $y$ are two positive real numbers such that $x y=4$, then the minimum value of $\left(\sqrt{x}+\frac{y^2}{2}\right)$ is

If the tangent and the normal drawn to the curve $x y^2+x^2 y=12$ at the point $(1,3)$ meet the X -axis in $T$ and $N$ respectively, then $T N=$

A man of 5 feet height is walking away from a light fixed at a height of 15 feet at the rate of of $K$ miles/hour. If the rate of increase of his shadow is $\frac{11}{5}$ feet $/ \mathrm{sec}$, then $K=($ Take 1 mile $=5280$ feet $)$