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-?

Step 1
Consider the given function
${\displaystyle f\left(t\right)=5t+5\cot \left(\frac{t}{2}\right)}$
Step 2
Find the first derivative
${\displaystyle f^{\prime }\left(t\right)=\frac{d}{dt}(5t+5\cot \left(\frac{t}{2}\right))}$
${\displaystyle =\frac{d}{dt}\left(5t\right)+\frac{d}{dt}\left(5\cot \left(\frac{t}{2}\right)\right)}$
${\displaystyle =5-\frac{5}{2}{\csc }^2\left(\frac{t}{2}\right)}$
Step 3
Set the derivative equal to 0 to find the critical numbers
f'(t)=0
${\displaystyle \Rightarrow 5-\frac{5}{2}{\csc }^2\left(\frac{t}{2}\right)=0}$
${\displaystyle \Rightarrow {\csc }^2\left(\frac{t}{2}\right)=2}$
${\displaystyle \Rightarrow \csc \left(\frac{t}{2}\right)=\pm \sqrt{2}}$
${\displaystyle \Rightarrow t=\frac{\pi }{2},\frac{3\pi }{2}f\text{or}[\frac{\pi }{4},\frac{7\pi }{4}]}$
Step 4
Evaluate f(t) at the endpoints and the obtained critical points.
${\displaystyle f\left(\frac{\pi }{4}\right)=5\left(\frac{\pi }{4}\right)+5\cot \left(\frac{\pi }{8}\right)\approx 16}$
${\displaystyle f\left(\frac{\pi }{2}\right)=5\left(\frac{\pi }{2}\right)+5\cot \left(\frac{\pi }{4}\right)\approx 12.85}$
${\displaystyle f\left(3\frac{\pi }{2}\right)=5\left(3\frac{\pi }{2}\right)+5\cot \left(3\frac{\pi }{4}\right)\approx 18.56}$
${\displaystyle f\left(7\frac{\pi }{4}\right)=5\left(7\frac{\pi }{4}\right)+5\cot \left(7\frac{\pi }{8}\right)\approx 15.42}$
Step 5
Therefore, the function f has
absolute minimum value 12.85
absolute maximum value 18.56