Question

find the maximum and minimum values of the function ( g(\theta)=4 \theta - 7 sin (\theta) ) on the interval ( left0, \frac{pi}{2}
ight
minimum value =
maximum value =

Explanation:

Step1: Find the derivative of the function

The derivative of \(g(\theta)=4\theta - 7\sin(\theta)\) is \(g'(\theta)=4 - 7\cos(\theta)\).

Step2: Find the critical points

Set \(g'(\theta) = 0\), so \(4-7\cos(\theta)=0\). Then \(\cos(\theta)=\frac{4}{7}\), and \(\theta=\cos^{- 1}(\frac{4}{7})\approx0.96\) (since \(\theta\in[0,\frac{\pi}{2}]\)).

Step3: Evaluate the function at critical points and endpoints

• At \(\theta = 0\): \(g(0)=4\times0-7\sin(0)=0\).
• At \(\theta=\cos^{-1}(\frac{4}{7})\): \(g(\cos^{-1}(\frac{4}{7})) = 4\cos^{-1}(\frac{4}{7})-7\times\frac{\sqrt{7^{2}-4^{2}}}{7}=4\cos^{-1}(\frac{4}{7})-\sqrt{33}\approx4\times0.96 - 5.74=3.84 - 5.74=-1.9\).
• At \(\theta=\frac{\pi}{2}\): \(g(\frac{\pi}{2})=4\times\frac{\pi}{2}-7\sin(\frac{\pi}{2}) = 2\pi-7\approx6.28 - 7=-0.72\).

Answer:

Minimum value = \(-1.9\)
Maximum value = \(0\)