The problem I'm in stuck is
For $\forall x_1, x_2\in\mathbb R$, prove that $$\frac{\sin(x_1)-\sin(x_2)}{x_1-x_2}$$ has neither maximum nor minimum.
I tried to fix $x_2$ to make a single variable function $$f(x):=\frac{\sin(x)-\sin(x_2)}{x-x_2}$$ and find the point $x_0$ such that $f'(x_0)=0$. I wanted to find some properties of $f(x_0)$, so that if $x_2$ varies, I wanted to show that $g(x_2):=f(x_0(x_2))$ has no maximum and minimum. One that I found is $$\frac{\sin(x_0)-\sin(x_2)}{x_0-x_2}=\cos(x_0)$$ so it somehow implies that $f(x_0)=\cos(x_0)$, which is nothing helpful to me. How can I make the direction from here? Thanks in advance.