I have $2$ questions in my homework that I can not figure out how to deal with:

$1.$ Let $(a,b,c)\in \mathbb{R}^3$, $a>1$ and $0<b \leq 1$. Prove that $a x=b \sin x+c$ has one solution.

For the first question I understand I need to use Intermediate value theorem, but how? $x \mapsto a x$ is continuous and $x \mapsto b \sin x$ is in range of $(-b,b)$. So there must be a one point of intersection but how do I prove it ?

$2.$ Let $f$ be a continuous function between $[a,b]$ and differentiable between $(a,b)$. Assume that $f^2(b)-f^2(a)=b^2-a^2$. Prove there exists $x\in(a,b)$ s.t. $f^′(x)f(x)-x=0$.

hint given: $g(x)=f^2(x)-x^2$

For the second question I have no idea how to even approach it.

Any help would be appreciated.