Let f : R rightarrow R and g : R rightarrow R | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $f(x+y) = f(x) + f(y) + f(x)f(y)$. Adding $1$ to both sides gives $1 + f(x+y) = (1 + f(x))(1 + f(y))$.Let $h(x) = 1 + f(x)$. Then $h(x+y) =

Vedclass · Accepted Answer

$A, B, D$

Vedclass · Answer

(B) Given $f(x+y) = f(x) + f(y) + f(x)f(y)$. Adding $1$ to both sides gives $1 + f(x+y) = (1 + f(x))(1 + f(y))$.Let $h(x) = 1 + f(x)$. Then $h(x+y) = h(x)h(y)$.Since $f(x) = xg(x)$ and $\lim_{x 	o 0} g(x) = 1$,we have $\lim_{x 	o 0} f(x) = 0$,so $h(0) = 1 + f(0) = 1$.For $h(x+y) = h(x)h(y)$,the solution is $h(x) = e^{cx}$.Thus $1 + f(x) = e^{cx}$,or $f(x) = e^{cx} - 1$.Given $f(x) = xg(x)$,we have $g(x) = \frac{e^{cx}-1}{x}$.$\lim_{x 	o 0} g(x) = \lim_{x 	o 0} \frac{e^{cx}-1}{x} = c$.Since $\lim_{x 	o 0} g(x) = 1$,we have $c = 1$.Therefore,$f(x) = e^x - 1$.$f'(x) = e^x$,which is defined for all $x \in R$,so $(A)$ is $TRUE$.$f'(0) = e^0 = 1$,so $(D)$ is $TRUE$.$f'(1) = e^1 = e 
eq 1$,so $(C)$ is $FALSE$.For $g(x) = \frac{e^x-1}{x}$ for $x 
eq 0$ and $g(0)=1$,$g$ is differentiable at all $x 
eq 0$. At $x=0$,$g'(0) = \lim_{h 	o 0} \frac{\frac{e^h-1}{h}-1}{h} = \lim_{h 	o 0} \frac{e^h-1-h}{h^2} = \frac{1}{2}$. Thus $g$ is differentiable everywhere,so $(B)$ is $TRUE$.

Similar Questions

Let a function $g:[0,4] \rightarrow R$ be defined as
$g(x) = \begin{cases} \max_{0 \leq t \leq x} \{t^3 - 6t^2 + 9t - 3\} & , 0 \leq x \leq 3 \\ 4 - x & , 3 < x \leq 4 \end{cases}$
Then the number of points in the interval $(0,4)$ where $g(x)$ is $NOT$ differentiable is $.....$

Let $f(x) = \begin{cases} (x - 1) \sin \frac{1}{x - 1}, & x \neq 1 \\ 0, & x = 1 \end{cases}$. Then which one of the following is true?

If the function $g(x)=\begin{cases} K \sqrt{x+1} &, 0 \leq x \leq 3 \\ mx+2 &, 3 < x \leq 5 \end{cases}$ is differentiable,then $K+m=$

The derivative of $y = 1 - |x|$ at $x = 0$ is

Let $f: R \rightarrow R$ be a function defined by $f(x) = \max \{x, x^2\}$. Let $S$ denote the set of all points in $R$ where $f$ is not differentiable. Then $S$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module