A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series

$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\ldots \ldots+f\left(\frac{39}{20}\right)$ is equal to ……. .

The largest interval lying in $\left( { – \frac{\pi }{2},\frac{\pi }{2}} \right)$ for which the function, $f\left( x \right) = {4^{ – {x^2}}} + {\cos ^{ – 1}}\left( {\frac{x}{2} – 1} \right) + \log \left( {\cos x} \right)$  is defined is

Suppose $f:[2,\;2] \to R$ is defined by $f(x) = \left\{ \begin{array}{l} – 1\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{for}}\; – 2 \le x \le 0\\x – 1\;\;\;\;\;{\rm{for}}\;0 \le x \le 2\end{array} \right.$, then $\{ x \in ( – 2,\;2):x \le 0$ and $f(|x|) = x\} = $

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree $3$ such that $\mathrm{f}(\mathrm{k})=-\frac{2}{\mathrm{k}}$ for $\mathrm{k}=2,3,4,5 .$ Then the value of $52-10 \mathrm{f}(10)$ is equal to :

Least integer in the range of $f(x)$=$\sqrt {(x + 4)(1 – x)}  – {\log _2}x$ is