0
$\begingroup$

It's my last question. Just give me advise how to start.

Q: Find all such functions $$f:\mathbb R\to \mathbb R,$$ for all real x, y, the equality $$f(yf(x))=x^2y^4$$

$\endgroup$
  • $\begingroup$ are the function continuous ? derivable ? $\endgroup$ – idm Nov 22 '14 at 9:27
  • 1
    $\begingroup$ Show that f(1) is not 0. $\endgroup$ – Paul Nov 22 '14 at 9:36
  • $\begingroup$ Function are contimuos. $\endgroup$ – Vlad9pa Nov 22 '14 at 9:36
1
$\begingroup$

$ f(yf(1))=y^4, $ so $f(1)\ne0$, so $f(y)=\left(\frac{y}{f(1)}\right)^4$. Now putting $y=1$ in last equation we get $f(1)=1$ and $f(y)=y^4$. Now it is easy too see that $f(y)=y^4$ cannot satisfy the equation $f(yf(x))=x^2y^4$

$\endgroup$
  • $\begingroup$ why is The first sentence true? $\endgroup$ – user139708 Nov 22 '14 at 9:49
  • $\begingroup$ Which part do you mean? $\endgroup$ – pointer Nov 22 '14 at 9:51
  • $\begingroup$ The first sentence of your answer. $\endgroup$ – user139708 Nov 22 '14 at 9:52
  • $\begingroup$ There are 3 formulas in the first sentence. Which of them don't you agree with? $\endgroup$ – pointer Nov 22 '14 at 9:56
  • $\begingroup$ The third one I dont understant. $\endgroup$ – user139708 Nov 22 '14 at 9:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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