Amit M Agarwal Solutions for Chapter: Continuity and Differentiability, Exercise 3: Target Exercises

Let $f\left(x\right)$ be a real function not identically zero in $Z$, such that $f\left(x+y^{2n+1}\right)=f\left(x\right)+{\left\{f\left(y\right)\right\}}^{2n+1}$ $n\in N$ and $x, y\in R$. If $f^{\text{'}}\left(0\right)\ge 0$, then $f^{\text{'}}\left(6\right)$ is equal to

Let $f\left(x+y\right)=f\left(x\right)·f\left(y\right)$ and $f\left(x\right)=1+x g\left(x\right)G\left(x\right)$, where $\underset{x\to 0}{\lim }g\left(x\right)=a$ and $\underset{x\to 0}{\lim }G\left(x\right)=b$. Then $f^{\text{'}}\left(x\right)=kf\left(x\right)$, where $k$ is equal to

If $f\left(x\right)=\left[{\log }_{e}x\right]+\sqrt{\left\{{\log }_{e}x\right\}}, x>1$, where $\left[\right]$ and $\left\{\right\}$ denote the greatest integer function and the fractional part function respectively, then

If $f\left(2+x\right)=f\left(-x\right)$ for all $x\in R$, then differentiability at $x=4$ implies differentiability at

The number of points of non-differentiability for $f\left(x\right)=\max \left\{\left|\left|x\right|-1\right|,\frac{1}{2}\right\}$ is

If $f\left(x\right)=x^{3}sgnx$, then

If $f\left(x\right)=\left\{\begin{array}{ll}\left|1-4x^2\right|,& 0\le x<1\\ \left[x^2-2x\right],& 1\le x<2\end{array}\right.$, where $\left[\right]$ denotes the greatest integer function, then $f\left(x\right)$ is

If $f\left(x\right)=\left\{\begin{array}{cl}x-1,& x<0\\ x^2-2x,& x\ge 0\end{array}\right.$, then