Mathematics

Problem

Define

h(x)=x\cot x

for

x

in the interval

(0,\pi)

. (a) Determine

h'(x)

and identify all critical numbers of

h

(0,\pi)

. (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,\pi)

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.

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