Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Find all differentiable functions $f \colon (0,\infty) \to \mathbb R$ for which there is a positive real number $k$ such that: $$ f(x) \cdot f'(k/x) = x, \qquad\text{for all }x > 0. $$

share|improve this question

1 Answer 1

up vote 4 down vote accepted

In equation $$ f(x)f'\left(\frac{k}{x}\right)=x $$ we make change of variables $k/x\to x$ then for all $x>0$ we get $$ f\left(\frac{k}{x}\right)f'(x)=\frac{k}{x} $$ Hence $$ \left(f(x)f\left(\frac{k}{x}\right)\right)'=f'(x)f\left(\frac{k}{x}\right)+f(x)f'\left(\frac{k}{x}\right)\frac{-k}{x^2}=\frac{k}{x}+x\frac{-k}{x^2}=0 $$ The rest is clear.

share|improve this answer
    
hmm. we get f(x) f(k/x) = C f(x) = x/f'(k/x) f'(k/x)/f(k/x) = C/x ln(f(k/x)) = Clnx + D f(k/x) = e^D x^C Is this correct? –  John Chang Nov 13 '12 at 7:55
    
@JohnChang You made two mistakes. The first: you must arrive at $f'(k/x)/f(k/x) = x/C$. The second: even if there were not the first misatke the step from $f'(k/x)/f(k/x) = C/x$ to $\ln(f(k/x)) = Clnx + D$ is incorrect. You need to make change $k/x\to x$ in $f'(k/x)/f(k/x) = C/x$. –  userNaN Nov 13 '12 at 8:17

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.