MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Define h(x)=xcotx for x in the interval (0,π).
(a) Determine h′(x) and identify all critical numbers of h in (0,π).
(b) Use appropriate tests to classify each critical number as yielding a local maximum or a local minimum for h. Show all supporting analysis.
(c) Evaluate h′′(x) and identify any points in (0,π) where the concavity of h changes. State each interval of concavity.
(d) Summarize your findings by listing all local extrema, their approximate numerical values (to three decimal places), and the intervals on which h is increasing or decreasing.