MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
Let
f(x)=(x−3)2/3x,x=3,
and consider the closed interval [−1,5].
(a) Determine all values of x in the interval [−1,5] at which f′(x)=0.
(b) Identify all values in [−1,5] where f′(x) is undefined while f itself is defined. Classify each as an interior point or an endpoint of a sub-interval of the domain.
(c) Construct a complete sign analysis for f′(x) on [−1,5], taking into account endpoints and any points where f is not defined.
(d) Using your work from part (c), determine the x-coordinates of every relative maximum and minimum of f on [−1,5], and justify each classification. Be sure to discuss the behavior near x=3 as part of your explanation.
(e) Decide whether the absolute maximum or absolute minimum value of f on [−1,5] occurs at a critical point from part (a), a point from part (b), or at an endpoint of [−1,5]. Support your answer with appropriate comparisons of function values.