Let $f;[a,b]\rightarrow \mathbb{R}$ be continuous on $[a,b]$ and differentiable n $(a,b)$.
We consider the points $A(a,f(a))$ and $B(b,f(b))$. There $c\in (a,b)$ such that the point $M(c,f(c))$ belongs to the chord $AB$.
Show that a tangent of the graph of $f$, that passes through $A$, exists. (it has to pass through $A$, $A$ is not an intersection).
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So we want to find a tangent line through $M$ that passes (and not intersects) $A$, or have I understood that wrong?
Is that line of the form $$y-f(a)=\frac{f(c)-f(a)}{c-a}(x-a)$$ ?
The given question does not require us the knowledge of $M$. Instead, it asks for a tangent line passing through $A$. Therefore, the equation for tangent line at $x=a$ should be like this:
$$ y-f(a)=f'(a)(x-a) $$
EDIT: I doubt that the tangent line only exists for certain kinds of function. If we let $f(x)=|x|$ and $a=0$, the tangent line will not exist at all.