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Context: Let g(x) be a twice differentiable function with a continuous second derivative and also satisfying the property that$\ x/2 \leq g(x) \leq x/2 + 1$ for each positive real number x. We let$\ f(x) = g(x) + \sin(x) $.

Question: Prove that there exist infinitely many real numbers c such that f(c) = (c+1)/2

Now I understand that I would have to use the Intermediate Value Theorem in order to prove that those real numbers c exist. I also understand that a possible solution requires me to formulate a new function h(x) using f(x) and$\ (c+1)/2 $. However, I don't really understand where my first step would be or how to create such a function in the first plays. If someone could point me in the right direction that would be amazing, thanks!

TeriyakiNinja
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1 Answers1

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The easiest way to use the Intermediate Value Theorem is to apply it to a function $h(x)$ to show that there is a point where the function is equal to $0$. That suggests looking for a function that will take the value $0$ at $c$ exactly when $f(c) = \frac{1}{2}(c+1)$. Well, that means trying $$h(x) = f(x) - \frac{1}{2}(x+1).$$

Showing that there are infinitely many points $c$ where $f(c)=\frac{1}{2}(c+1)$ is the same as showing there are infinitely many points $c$ where $h(c)=0$. To that end, you want to show that you can guarantee that $h(c)$ is (i) continuous; and (ii) changes sign infinitely many times.

Arturo Magidin
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