Amit M Agarwal Solutions for Chapter: Functions, Exercise 10: Proficiency in 'Functions' Exercise 2

Let $f\left(x\right)=x^2+3x-3, x\ge 0$. $n$ points $x_1, x_2,\dots .,x_n$ are chosen on the $x-$axis such that

$\left(\mathrm{i}\right)$ $\frac{1}{n}\sum _{i=1}^n{f}^{-1}\left(x_i\right)=f\left(\frac{1}{n}\sum _{i=1}^n{x}_i\right)$

$\left(\mathrm{ii}\right)$ $\sum _{i=1}^n{f}^{-1}\left(x_i\right)=\sum _{i=1}^n{x}_i$, where $f^{-1}$ denotes the inverse of $f$. Find the $\mathrm{AM}$ of $x_i^{\text{'}}s$.

Let $f\left(x\right)=x^2-2x, x\in R$ and $g\left(x\right)=f\left(f\left(x\right)-1\right)+f\left(5-f\left(x\right)\right),$ show that $g\left(x\right)\ge 0,\forall x\in R$.

If $f$ is a polynomial function satisfying $2+f\left(x\right)·f\left(y\right)=f\left(x\right)+f\left(y\right)+f\left(xy\right)\forall x, y\in R$ and if $f\left(2\right)=5$, then find $f\left(f\left(2\right)\right)$.

$a+b+c=abc, a, b$ and $c\in R^{+}$, then prove that $a+b+c\ge 3\sqrt{3}.$

Consider the function $f\left(x\right)=\left\{\begin{array}{cc}x-\left[x\right]-\frac{1}{2},& \mathrm{i}\mathrm{f} x\notin I\\ 0,& \mathrm{i}\mathrm{f} x\in I\end{array}\right.$, where [.] denotes the greatest integral function and $I$ is the set of integers. Then, find $g\left(x\right)$ where $g\left(x\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{x^2, f\left(x\right), \left|x\right|\right\};-2\le x\le 2.$

If $f\left(x\right)$ be continuous function in $\left[0, 2\mathrm{\pi }\right]$ and $f\left(0\right)=f\left(2\mathrm{\pi }\right)$, then prove that there exists a point $c\in \left(0, \mathrm{\pi }\right)$ such that $f\left(c\right)=f\left(c+\mathrm{\pi }\right)$.

Let $g\left(t\right)=\left|t-1\right|-\left|t\right|+\left|t+1\right|,\forall t\in R$. Then, find $f\left(x\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{g\left(t\right):-\frac{3}{2}\le t\le x\right\},\forall x\in \left(\frac{-3}{2}, \infty \right)$.

Find the integral solution for $n_1{n}_2=2n_1-n_2$, where $n_1, n_2\in$ integer.