We began class with challenge problem presentations. We introduced Theorem 5.2.5 (Chain Rule) and went over some of its implications. One thing that is important to note about the usage of Chain Rule is that we can control the inputs of a function, specifically so that they are not equal, but we cannot do this for the outputs of functions. We also noted that the differentiability of functions used in Chain Rule tells us that the functions are continuous at those differentiable points. Next, we introduced Theorem 5.2.6 (Interior Extremum Theorem) and went through the outline of its proof. The proof involved picking points on either side of the minimum or maximum and then showing that the derivative at that point must be 0 with inequalities. After that, we went over Darboux’s Theorem which can be generalized to the case that if a function has both a positive and negative derivative and two distinct points, then the derivative must be 0 somewhere between these points. Lastly, we discussed the Generalized Mean Value Theorem which is an application of the Mean Value Theorem. In a broad sense, the theorem says that for a secant line of a continuous function, there is a point between the endpoints of the secant line such that the tangent line at that point has a slope equal to that of the secant line.