MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Let h(x)=x1/3(x−2), where h is defined for all real x.
(a) Find h′(x). Show all algebraic steps in your derivative calculation.
(b) Determine every real number c at which h has a critical point. Be sure to consider all ways in which the derivative may fail to be nonzero.
(c) For each critical point found in part (b), classify it as a local maximum, a local minimum, or neither. Use a first–derivative test, a second–derivative test, or another fully justified method of your choice. Clearly indicate which test you are using.
(d) Summarize how your work illustrates the relationship between critical points and local extrema for this function.