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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.

Axel
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1 Answers1

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As you made the effort of editing your post and include some of your thoughts, I will give you some hints:

First question: Let $f : x \mapsto ax-b\sin(x)$, notice that $f$ is strictly increasing on $\mathbb{R}$ by showing $f'>0$. Now what is $\displaystyle \lim_{x \to -\infty} f(x)$ and $ \displaystyle \lim_{x \to \infty} f(x)$? Can you finish this using the intermediate value theorem?

Second question: using the hint given, $g$ is differentiable on $(a,b)$ continuous on $[a,b]$ and its derivative is given by $g'(x)=2f(x)f'(x)-2x$. Now the assumption tells you that $g(a)=g(b)$ so try to use Rolle's theorem.

Axel
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