Find the absolute maximum and absolute minimum values of f on the given interval. f(t)=5t+5cot(t/2), [pi/4, 7pi/4] absolute minimum value-? absolute maximum value-?

Falak Kinney 2020-10-28 Answered
Find the absolute maximum and absolute minimum values of f on the given interval.
f(t)=5t+5cot(t2),[π4,7π4]
absolute minimum value-?
absolute maximum value-?
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Expert Answer

Margot Mill
Answered 2020-10-29 Author has 106 answers
Step 1
Consider the given function
f(t)=5t+5cot(t2)
Step 2
Find the first derivative
f(t)=ddt(5t+5cot(t2))
=ddt(5t)+ddt(5cot(t2))
=552csc2(t2)
Step 3
Set the derivative equal to 0 to find the critical numbers
f'(t)=0
552csc2(t2)=0
csc2(t2)=2
csc(t2)=±2
t=π2,3π2for[π4,7π4]
Step 4
Evaluate f(t) at the endpoints and the obtained critical points.
f(π4)=5(π4)+5cot(π8)16
f(π2)=5(π2)+5cot(π4)12.85
f(3π2)=5(3π2)+5cot(3π4)18.56
f(7π4)=5(7π4)+5cot(7π8)15.42
Step 5
Therefore, the function f has
absolute minimum value 12.85
absolute maximum value 18.56
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