0
$\begingroup$

I want to show that the function $$\frac{3}{2}x-6-\frac{1}{2}\sin(2x)$$ has one unique real root.

By looking at the graph, it is obvious that there is only one root. But how can I show that this is unique real root by a formal mathematical proof?

$\endgroup$
5
$\begingroup$

Use the graph to find points of opposite sign for the intermediate value theorem.

Use the derivative to prove strict monotonicity.

$\endgroup$
1
$\begingroup$

Rolle's theorem is enough. It is quite trivial that a root must exist by continuity, since $3x-12$ is unbounded while $\sin(2x)\in[-1,1]$. On the other hand, by assuming that $3x-12-\sin(2x)$ has two roots $a,b$, then the derivative $3-2\cos(2x)$ must vanish for some $x\in(a,b)$. That cannot happen, since $\cos(2x)\in[-1,1]$, too.

$\endgroup$

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.