# Show that a tangent of the graph of $f$, that passes through $A$, exists.

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).



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.