Question

find the absolute maximum and absolute minimum values of ( f ) on the given interval.

( f(t)=5 t+5 cot left(\frac{t}{2}
ight), quadleft\frac{pi}{4}, \frac{7 pi}{4}
ight )

absolute minimum value

absolute maximum value

resources

Explanation:

Step1: Find the derivative of \(f(t)\)

Use the sum rule \((u + v)^\prime=u^\prime+v^\prime\) and the chain - rule \((\cot u)^\prime=-\csc^{2}u\cdot u^\prime\).
If \(f(t)=5t + 5\cot(\frac{t}{2})\), then \(f^\prime(t)=5+5\times(-\csc^{2}(\frac{t}{2}))\times\frac{1}{2}=5-\frac{5}{2}\csc^{2}(\frac{t}{2})\).
Set \(f^\prime(t) = 0\), so \(5-\frac{5}{2}\csc^{2}(\frac{t}{2})=0\).

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Since \(t\in[\frac{\pi}{4},\frac{7\pi}{4}]\), then \(\frac{t}{2}\in[\frac{\pi}{8},\frac{7\pi}{8}]\) and \(\csc(\frac{t}{2})=\sqrt{2}\) (because \(\csc x=\frac{1}{\sin x}\gt0\) for \(x\in(0,\pi)\) in the given interval of \(\frac{t}{2}\)).
\(\sin(\frac{t}{2})=\frac{1}{\sqrt{2}}\), so \(\frac{t}{2}=\frac{\pi}{4}\) or \(\frac{3\pi}{4}\), and \(t = \frac{\pi}{2}\) or \(t=\frac{3\pi}{2}\).

Step2: Evaluate \(f(t)\) at critical points and endpoints

• When \(t=\frac{\pi}{4}\):

\(f(\frac{\pi}{4})=5\times\frac{\pi}{4}+5\cot(\frac{\pi}{8})\). Since \(\cot(\frac{\pi}{8})=\sqrt{2}+1\), then \(f(\frac{\pi}{4})=\frac{5\pi}{4}+5(\sqrt{2} + 1)\approx\frac{5\times3.14}{4}+5\times(1.414 + 1)\approx3.925+12.07\approx15.995\).

• When \(t=\frac{\pi}{2}\):

\(f(\frac{\pi}{2})=5\times\frac{\pi}{2}+5\cot(\frac{\pi}{4})=\frac{5\pi}{2}+5\times1\approx\frac{5\times3.14}{2}+5=7.85 + 5=12.85\).

• When \(t=\frac{3\pi}{2}\):

\(f(\frac{3\pi}{2})=5\times\frac{3\pi}{2}+5\cot(\frac{3\pi}{4})=\frac{15\pi}{2}+5\times(-1)\approx\frac{15\times3.14}{2}-5=23.55-5 = 18.55\).

• When \(t=\frac{7\pi}{4}\):

\(f(\frac{7\pi}{4})=5\times\frac{7\pi}{4}+5\cot(\frac{7\pi}{8})\). Since \(\cot(\frac{7\pi}{8})=-\cot(\frac{\pi}{8})=-(\sqrt{2}+1)\), then \(f(\frac{7\pi}{4})=\frac{35\pi}{4}-5(\sqrt{2}+1)\approx\frac{35\times3.14}{4}-12.07\approx27.475-12.07\approx15.405\).

Answer:

absolute minimum value: \(12.85\) (when \(t = \frac{\pi}{2}\))
absolute maximum value: \(18.55\) (when \(t=\frac{3\pi}{2}\))