Points A and B are on the graph of function g; the chord joining those points is shown (together with its slope). Is there a point P on the blue portion of that graph (i.e., the part between A and B) where the tangent at P is parallel to the AB chord?
Point C on the x-axis can be moved within the interval below the part of the graph between A and B. An arrow points from C to point P on the graph and the tangent at P is shown. If slope of that tangent is sufficiently close to the AB chord's slope, then key parts of the figure change color: red to green.

Geometric idea: the dashed line, "parallel to chord AB", goes through a point which can be moved freely.

Can they, point and dashed line, be moved so that the dashed line meets the blue curve at a single point.

If so, is it plausible that the dashed line is tangent there to the curve?

On the other hand, is there a curve (and choice of A and B on it) so that "single-point-contact" by the dashed line clearly does not involve tangency?

Hint: investigate the effect of a key condition for the Mean Value Theorem by specifying a function that is not differentiable somewhere (for instance, absolute value of x).
Note: slider "tol" adjusts the tolerance for checking the match of slopes.