Properties of Differentiable Functions

Let $f,g : \mathrm{ℝ}\to \mathrm{ℝ}$ be such that $f\left(x\right)=x^{k}g\left(x\right)$ where $k$ is a positive integer, Suppose $g$ is continuous at $x=0$. Then

A function $f:R\to R$ satisfies the relation $f(x+y)=f\left(x\right)·f\left(y\right), \forall x, y\in R$ and $f\left(x\right)\ne 0, \forall x\in R.$ If $\text{'}f\text{'}$ is differentiable at $x=0, f^{\text{'}}\left(0\right)=4$ and $f\left(6\right)=3,$ then $f^{\text{'}}\left(6\right)= \_\_\_\_\_$

Let $g\left(\frac{x+y}{2}\right)=\frac{g\left(x\right)+g\left(y\right)}{2}, \forall x\in R, y\in R$. If $g^{\text{'}}\left(0\right)=-1$ and $g\left(0\right)=1$, then $g\left(x\right)$ is $a$

Let $f:\left(-1,1\right)\to R$ be a differentiable function satisfying ${\left(f^{\text{'}}\left(x\right)\right)}^4=16{\left(f\left(x\right)\right)}^2$ for all $x\in \left(-1,1\right)$ $f\left(0\right)=0$. The number of such functions is

Evaluate the summation $\mathrm{f}\left(\frac{1}{2015}\right)+\mathrm{f}\left(\frac{2}{2015}\right)+\mathrm{f}\left(\frac{3}{2015}\right)+\dots \dots +\mathrm{f}\left(\frac{4029}{2015}\right)$ , given $\mathrm{f}\left(\mathrm{x}\right)=\frac{9^{\mathrm{x}}}{9^{\mathrm{x}}+9}$.

If $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)\cdot \mathrm{f(}y) \forall x, y\in R \& f\left(0\right)=0$, then

${\int }_{-51}^{51}\frac{dx}{3+f\left(x\right)}$ has the value equal to :

$f$ is a real valued function from $R$ to $R$such that $f\left(x\right)+f(-x)=2,$then $\underset{1-x}{\overset{1+x}{\int }}{f}^{-1}\left(t\right)dt=$

If $f(x+f\left(y\right))=f\left(x\right)+y \forall x,y\in R$ and $f\left(0\right)=1$, then $\underset{0}{\overset{10}{\int }}f(10-x)dx$ is equal to

Consider the function $f\left(x\right)=\min \left\{\left|x^2-4\right|,\mathrm{}\left|x^2-1\right|\right\}$, then the number of points where $f\left(x\right)$ is non-differentiable is/are

Let $f\left(x\right)$ be a non-negative differentiable function on $[0,\infty )$ such that $f\left(0\right)=0$ and $f^{\text{'}}\left(x\right)\le 2f\left(x\right)$ for all $x>0.$ Then, on $[0,\infty )$

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x}{\sqrt{1+x^2}}$ then $\left(fofof\right) \left(x\right)$ is

If $\frac{3x+4}{{\left(x+1\right)}^2 \left(x-1\right)}=\frac{A}{\left(x-1\right)}+\frac{B}{\left(x+1\right)}+\frac{C}{{\left(x+1\right)}^2}$ , then $A=$

Let $f\left(\frac{x+y}{2}\right)=\frac{f\left(x\right)+f\left(y\right)}{2}$ for real values of $x$ and $y$ . If $f^{\text{'}}\left(0\right)$ exist and equals $-1$ and $f\left(0\right)=1$, then $f^{\text{'}}\left(2\right)$ is equal to-

Let  f : R $\to$ R be a function such that $\text{f}\left(\frac{\text{x}+\text{y}}{3}\right)=\frac{\text{f}\left(\text{x}\right)+\text{f}\left(\text{y}\right)}{3}\text{, }\text{f}\left(0\right)=0\text{ and }{\text{f}}^{\prime }\left(0\right)=3$ then f(x) is :

Which one of the following is not true always?

If $\left[x\right]$ denotes the integral part of $x$ and $f\left(x\right)=[n+p\sin x]$, $0<x<\mathrm{\pi }\text{,} n\in I$ and $p$ is a prime number, then the number of points, where $f\left(x\right)$ is not differentiable, is

Let $f:\left(-1, 1\right)\to R$ be a differentiable function with $f\left(0\right)=-1$ and $f^{\text{'}}\left(0\right)=1$, $g\left(x\right)={\left\{f\left(2f\left(x\right)+2\right)\right\}}^2$ . Then $g^{\text{'}}\left(0\right)$