Answer
The absolute maximum is $f(\frac{3\pi}{2})= 3.71$
The absolute minimum is $f(\frac{\pi}{2})= 2.57$
Work Step by Step
$f(t) =t + cot(\frac{t}{2})$
We can find the points where $f'(t) = 0$:
$f'(t) = 1 - \frac{1}{2~sin^2~(t/2)} = 0$
$1 - \frac{1}{2~sin^2~(t/2)} = 0$
$\frac{1}{2~sin^2~(t/2)} = 1$
$2~sin^2~(\frac{t}{2}) = 1$
$sin^2~(\frac{t}{2}) = \frac{1}{2}$
$sin~(\frac{t}{2}) = \pm \frac{1}{\sqrt{2}}$
$\frac{t}{2} = \frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4},...$
$t = \frac{\pi}{2},\frac{3\pi}{2}$
We can verify the values of these two points and the endpoints of the interval:
$f(\frac{\pi}{4}) = (\frac{\pi}{4})+cot(\frac{\pi}{8}) = 3.20$
$f(\frac{\pi}{2}) = (\frac{\pi}{2})+cot(\frac{\pi}{4}) = 2.57$
$f(\frac{3\pi}{2}) = (\frac{3\pi}{2})+cot(\frac{3\pi}{4}) = 3.71$
$f(\frac{7\pi}{4}) = (\frac{7\pi}{4})+cot(\frac{7\pi}{8}) = 3.08$
The absolute maximum is $f(\frac{3\pi}{2})= 3.71$
The absolute minimum is $f(\frac{\pi}{2})= 2.57$
