Remember Theresa, the girl who loves tangents? Well, Theresa is at it again! She has drawn tangents to a function and then erased the function.

1. Trace her diagram on your paper. Using the tangents, sketch a possible function.

   Hint (a):

   Make sure your sketch of $g\left(x\right)$ goes through all the coordinate points in order AND has the indicated slope at each point.

2. Is your function differentiable everywhere? Justify your answer.

   Hint (b):

   Recall that a function is NOT differentiable at cusps, endpoints, jumps, holes and vertical tangents. In other words, there can be no sudden changes in slope, infinite slopes or breaks in continuity.

3. Could any of the points shown be the location of a local maximum or minimum? Justify your ideas.

   Hint (c):

   Recall that a local max or local min COULD exist in one of two places: where the derivative (slope) $= 0$ or where the derivative does not exist.

   Hint (c):

   But, just because $g^\prime(x) = 0$ or $g^\prime(x) = \text{DNE}$, does not mean that that MUST be the location of a local max or local min.