# explore the MVT (Mean Value Theorem)

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?

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