I have the following question:
"Under what condition does the following equation have a real solution? $$ b x=2\sinh(a x), $$ where $a$ and $b$ are given real constants."
I can simply think of a solution by intersecting the line $y_{1}=bx$ with the curve of $y_{2}=2\sinh(ax)$. For them to intersect, the slope of the line ($b$) has to be larger than the slope of the tangent of $y_{2}$ at $x=0$, which gives the condition: $b\geq 2a$. Is this correct? And would you say this is the answer the question is looking for?
The question is still puzzling to me, as it asks for the condition of a real solution (not just a 'solution'). Any ideas or suggestions about this? How different would things be for a complex solution? Or is the question just badly phrased?