If f(x) is a differentiable function such tha | vedclass

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

Question

(B) Given the functional equation $f(xy) = f(x) + f(y)$.Setting $x = 1$ and $y = 1$,we get $f(1) = f(1) + f(1)$,which implies $f(1) = 0$.Now,different

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(B) Given the functional equation $f(xy) = f(x) + f(y)$.Setting $x = 1$ and $y = 1$,we get $f(1) = f(1) + f(1)$,which implies $f(1) = 0$.Now,differentiate the given equation with respect to $y$ keeping $x$ constant:$\frac{d}{dy} f(xy) = \frac{d}{dy} (f(x) + f(y))$$f'(xy) \cdot x = f'(y)$.Setting $y = 1$,we get $x \cdot f'(x) = f'(1)$.Let $f'(1) = k$,then $f'(x) = \frac{k}{x}$.Integrating both sides with respect to $x$,we get $f(x) = k \ln(x) + C$.Since $f(1) = 0$,we have $k \ln(1) + C = 0$,which implies $C = 0$.Thus,$f(x) = k \ln(x)$.Now,calculate $f(e) + f(1/e)$:$f(e) = k \ln(e) = k(1) = k$.$f(1/e) = k \ln(1/e) = k \ln(e^{-1}) = -k \ln(e) = -k$.Therefore,$f(e) + f(1/e) = k + (-k) = 0$.

If $f(x)$ is a differentiable function such that $f(xy) = f(x) + f(y)$ for all $x, y > 0$,then $f(e) + f(1/e) = ?$

Similar Questions

If $f(x)$ is a quadratic function such that $f(x) f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right)$,then $\sqrt{f\left(\frac{2}{3}\right) + f\left(\frac{3}{2}\right)} = $

If $f(x) = x - \frac{1}{x}$, $x \neq 0$, then $3f(x) =$

The function $f(x) = x - [x]$,where $[x]$ denotes the greatest integer function,is:

Let $f: R \rightarrow R$ be a twice differentiable function such that $(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$ for all $x, y \in R$. If $f'(0) = \frac{1}{2}$,then the value of $24f''\left(\frac{5\pi}{3}\right)$ is:

$A$ function $f: R \rightarrow R$ satisfies the relation $f(x+y)=f(x) \cdot f(y), \forall x, y \in R$ and $f(x) \neq 0, \forall x \in R$. If $f$ is differentiable at $x=0$,$f^{\prime}(0)=4$,and $f(6)=3$,then $f^{\prime}(6)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module