Amit M Agarwal Solutions for Chapter: Continuity and Differentiability, Exercise 6: Proficiency in 'Continuity and Differentiability' Exercise 2

A function $f:R\to R$ satisfies the equation $f\left(x+y\right)=f\left(x\right)f\left(y\right)$ for all $x,y$ in $R$ and $f\left(x\right)\ne 0$, for any $x$ in $R$. Let the function be differentiable at $x=0$ and $f\text{'}\left(0\right)=2$. Show that $f\text{'}\left(x\right)=2f\left(x\right)$, for all $x$ in $R$. Hence, determine $f\left(x\right)$.

Suppose the function $f$ satisfies the following two conditions for all $x, y\in R$

$\left(\mathrm{i}\right)$  $f\left(x+y\right)=f\left(x\right)f\left(y\right)$                   $\left(\mathrm{ii}\right)$ $f\left(x\right)=1+xg\left(x\right)$, where $\underset{x\to 0}{\lim }g\left(x\right)=1$

Prove that the derivative $f^{\text{'}}\left(x\right)$, exists and $f^{\text{'}}\left(x\right)=f\left(x\right)$.

Let, $f\left(xy\right)=f\left(x\right)f\left(y\right)$ and $f$ is differentiable at $x=1$, such that $f\text{'}\left(1\right)=1$ also $f\left(1\right)\ne 0$, then show that $f$ is differentiable for all $x\ne 0$. Hence, determine $f\left(x\right)$.

A function $f:R\to R$, where $R$ is a set of real numbers, satisfying the equation $f\left(\frac{x+y}{3}\right)=\frac{f\left(x\right)+f\left(y\right)+f\left(0\right)}{3}$ for all $x,y$ in $R$. If the function is differentiable at $x=0$, then show that it is differentiable for all $x$ in $R$.

Let, $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+2xy-1$, for all real $x,y$ and $f\left(x\right)$ be differentiable function. If $f^{\text{'}}\left(0\right)=\cos \alpha$, then prove that $f\left(x\right)>0 \forall x\in R$.

If $f\left(x\right)$ is a real-valued function not identically equal to zero, such that $f\left(x+y^n\right)=f\left(x\right)+f^n\left(y\right)$, $x,y\in R$ and $n$ is natural number $>1$ and $f\text{'}\left(0\right)\ge 0$, then find the values of $f\left(5\right)$ and $f\text{'}\left(10\right)$.

Find the area of the region bounded by $y=f\left(x\right), y=\left|g\left(x\right)\right|$ and the lines $x=0, x=2$ where $f$ and $g$ satisfying $f\left(x+y\right)=f\left(x\right)+f\left(y\right)-8xy \forall x,y\in R$ and $g\left(x+y\right)=g\left(x\right)+g\left(y\right)+3xy\left(x+y\right) \forall x,y\in R$ also $f\text{'}\left(0\right)=8, g\text{'}\left(0\right)=-4$.

Let, $f:R\to R$, is a real-valued differentiable function $\forall x,y\in R$ such that $\left|f\left(x\right)-f\left(y\right)\right|\le {\left|x-y\right|}^3$. Prove that $h\left(x\right)=\int f\left(x\right)dx$ is continuous function of $x,\forall x\in R$.