Give an appropriate answer.Show that the function r(?) = 3 cot ? + 
+ 7 has exactly one zero on the interval (0, ?).
What will be an ideal response?
The function r(?) is continuous on the open interval (0, ?). Also, r(?) approaches ? as ? approaches 0 from the right, and r(?) approaches -? as ? approaches ? from the left. Since r(?) is continuous and changes sign along the interval, it must have at least one root on the interval.
The first derivative of r(?) is r'(?) = -3 csc2 ? - 
, which is everywhere negative on (0, ?). Thus, r(?) has a single root on (0, ?).
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