Q:

Find the number of real solutions of $x^2+e^x=\sin x$

Find the number of real solutions of:
$e^x+x^2= \sin x$
Graphically, I am unable to derive any solutions. Also, how could I estimate the real solutions?

A:

There are no solutions. Intuitively, this graph shows you why there are no solutions.
For the analytic proof there are no solutions, Prove that for $x\notin[-2,0]$, the function $e^x+x^2$ is larger than $1$, while on $[-2,0]$, it is larger than $0$.
More hints: Showing that $e^x+x^2 > 0$ should be trivial. The fact that $e^x + x^2>x^2$ should prove that $e^x+x^2>1$ for $x<-2$, and a similar expression will take care of $x>0$.

