# 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-?

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