Derivative Test

  Problem 1. Consider the function f(x) = x 5/4 − 8x 1/4 (a) Find f 0 (x). (b) Find the critical points of f(x) and state whether each one is a stationary point or a singular point. Hint: use a common denominator to write the function in the form Ax+B Cx1/D for some appropriate constants. 1 2 CALCULUS I, FALL 2022 MATH 1210-090 INSTRUCTOR: WILL FELDMAN (c) Use the First Derivative Test to classify each critical point found in (b) as a local minimum, local maximum, or neither. Problem 2. A manufacturer is making cylindrical cans with radius r and height h. Corporate says the volume of the cans must be 500 mL (i.e. cm3 ) exactly. Corporate also says the base of the can should be made out of a “smart wifi-enabled” material that tracks consumer can use patterns. This material costs 3 times as much as the material used for the sides and top of the can. What are the dimensions r and h of the can that minimize the total cost.