I'm struggling with the following problem:

For every $a\in\Bbb{R}$ find the number of solutions (where $x\in\Bbb{R}$) for $e^{-|x|}(x-1)^3=a$

After some transformations we get $(x-1)^3-ae^{|x|}=0$. Now I had an idea of finding critical points of the derivative of this function and examining its extremums, but the derivative is still going to have $e$ at some point so it leads to a similair problem. So, how to examine all $a$'s at once? Are the derivatives the way to go?


Consider the graph of the function

$$ y(x)=e^{-|x|}(x-1)^3 $$

The solution to equation in question is intersection point of graph of $y$ and graph of $y=a$. Now that greatly simplifies the problem.

Notice also that as $x\to \pm \infty$, we have $y\to 0$ as $y=O(e^{-|x|})$ (exponential functions dominate polynomials for large $x$).

$y=0$ if and only if $x=1$ and is $y>0$ for $x>1$, $y<0$ for $x<1$.

Can you imagine sketch of $y$ now? Then only thing you need to find is critical points of $y$, that is, where the maxima/minima occurs. Then you can easily classify number of solutions according to value of $a$ (and for this use of derivatives is probably best method).


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.