Define g(x)={sin⁡x,0≤x<π21,x=π2cos⁡x,π2<x≤πg(x)=\begin{cases} \sin x, & 0\le x<\tfrac{\pi}{2}\\ 1, & x=\tfrac{\pi}{2}\\ \cos x, & \tfrac{\pi}{2}<x\le \pi \end{cases}…

Mathematics

Problem

Define g(x)={sin⁡x,0≤x<π21,x=π2cos⁡x,π2<x≤πg(x)=\begin{cases} \sin x, & 0\le x<\tfrac{\pi}{2}\\ 1, & x=\tfrac{\pi}{2}\\ \cos x, & \tfrac{\pi}{2}<x\le \pi \end{cases} Does there exist a value cc in [0,π][0,\pi] for which g(c)=0.8g(c)=0.8? Answer "yes" or "no" and name an interval on which your conclusion is guaranteed.

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