Prove that for all $c \in [0,\infty)$ there exists $x \in \mathbb{R}$ such that $xe^x=c$ by the intermediate value theorem
Solution 1:
hint
If $ c=0$, we will take $ x=0$.
So, let us assume that $ c>0$.
the function $ f : x\mapsto xe^x $ satisfies $$\lim_{x\to 0^+}f(x)=0\;;\;\lim_{x\to\infty}f(x)=+\infty $$
So, there exist $ \delta>0 $ and $ A>0 $ such that $$a\le \delta \implies f(a)<\frac c2$$ and $$b\ge A \implies f(b)>2c$$
Now, apply IVT to $ g:x\mapsto f(x)-c $
at the intervall $[a,b] $.