Let $f(x) = (1 + {b^2}){x^2} + 2bx + 1$ and $m(b)$ the minimum value of $f(x)$ for a given $b$. As $b$ varies, the range of $m(b)$ is

Let $f : R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to.

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a function which satisfies $\mathrm{f}(\mathrm{x}+\mathrm{y})=\mathrm{f}(\mathrm{x})+\mathrm{f}(\mathrm{y}) \forall \mathrm{x}, \mathrm{y} \in \mathrm{R} .$ If $\mathrm{f}(1)=2$ and $g(n)=\sum \limits_{k=1}^{(n-1)} f(k), n \in N$ then the value of $n,$ for which $\mathrm{g}(\mathrm{n})=20,$ is 

The range of function $f : R \rightarrow  R$, $f(x) = \frac{{{{(x\, + \,1)}^4}}}{{{x^4} + \,1}}$ is

If $h\left( x \right) = \left[ {\ln \frac{x}{e}} \right] + \left[ {\ln \frac{e}{x}} \right]$ ,where [.] denotes greatest integer function, then which of the following is false ?

Period of $f(x) = nx + n - [nx + n]$, $n \in N$

where [ ] denotes the greatest integer function is :