1. DISCUSSION QUESTIONS Work together in your group to solve the following problems. Then go back later and solve them again by yourself before writing and uploading your final answer. (1) Give an example of a differentiable function f(x) and a critical point of f(x) which is neither a maximum nor a minimum. Specify where the critical point occurs and why it is neither a maximum nor a minimum. (2) Give an example of a continuous function f(x) with a local minimum at a point where f(x) is not differentiable. Explain why the function is not differentiable at this local minimum. (3) Give an example of a continuous function f(x) and a point p where f(x) is not differentiable but which is not a local max or min. Specify which point and explain why the function is not differentiable there. (4) Give an example of a function f(x) which is concave up on [2, 5]. Then graph y = f(x) and the secant line to y = f(x) through (2, f(2)) and (5, f(5)). Now, give an example of a function g(x) which is concave down on [2,5]. Then graph y= g(x) and the secant line to y = g(x) through (2, g(2)) and (5, g(5)). What do you notice about the secant lines versus the graphs in the two cases? Formulate a general principle about secant lines and concavity. (5) The term "inflection point" is used fairly often in colloquial speech. Use Google or some other search to find three times the word "inflection point" was used in the popular press (in different settings). (For example, if you Google "Kamala Harris inflection point" you'll find one that got a lot of attention. Or try Google news.) Give your examples (and say where they came from), and say a sentence or two about how each usage compares with the mathematical meaning of the word. 1