Question

find the maximum and minimum values of the function ( g(\theta)=5\theta - 6sin(\theta) ) on the interval ( 0,pi ) minimum value = maximum value =

Explanation:

Step1: Find the derivative of \(g(\theta)\)

The derivative of \(g(\theta)=5\theta - 6\sin(\theta)\) is \(g'(\theta)=5 - 6\cos(\theta)\) using the rules \((x^n)'=nx^{n - 1}\) and \((\sin x)'=\cos x\).

Step2: Find the critical points

Set \(g'(\theta)=0\), so \(5 - 6\cos(\theta)=0\). Then \(\cos(\theta)=\frac{5}{6}\). Using \(\theta=\cos^{-1}(\frac{5}{6})\approx0.585\) (since \(\theta\in[0,\pi]\)).

Step3: Evaluate \(g(\theta)\) at critical points and endpoints

• At \(\theta = 0\): \(g(0)=5\times0-6\sin(0)=0\).
• At \(\theta=\cos^{-1}(\frac{5}{6})\): \(g(\cos^{-1}(\frac{5}{6})) = 5\cos^{-1}(\frac{5}{6})-6\times\frac{\sqrt{6^{2}-5^{2}}}{6}=5\cos^{-1}(\frac{5}{6})-\sqrt{11}\approx5\times0.585 - 3.317=2.925-3.317=- 0.392\).
• At \(\theta=\pi\): \(g(\pi)=5\pi-6\sin(\pi)=5\pi\approx15.71\).

Answer:

Minimum value = \(-0.392\)
Maximum value = \(5\pi\approx15.71\)