A function is continuous on the closed interval and differentiable on except at
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function g is continuous on the closed interval [−4,4] and differentiable on (−4,4) except at x=0, where its graph has a sharp corner. At that point the derivative of g does not exist, but g itself is defined and continuous.
Answer the following questions about locating the absolute (global) extrema of g on [−4,4].
(a) State the complete definition of a critical point of a function on an open interval.
(b) Describe the full collection of x–values that must be examined when searching for the absolute maximum and minimum of a continuous function on a closed interval, and explain why each type of point is included.
(c) Give a detailed justification for why a point such as x=0, where g′(0) does not exist but g is continuous, must be considered as a potential location of an absolute maximum or minimum.
(d) For the specific function g(x)=∣x∣3/2+1, defined on [−4,4], determine whether x=0 is an actual location of an absolute maximum, an absolute minimum, or neither. Justify your conclusion without computing any derivatives at x=0.