Let f:[0,)[0,3] be defined by f(x)=cases t:0 t x, & 0 x

Question

Let $f:[0,\infty)	o[0,3]$ be defined by $f(x)=\begin{cases}\max\{\sin t:0\le t\le x\}, & 0\le x<\pi\2+\cos x, & x\ge\pi\end{cases}$. Then which of the following is true?

Accepted Answer

On $[0,\pi/2]$, $\max\{\sin t:t\le x\}=\sin x$ (increasing); on $[\pi/2,\pi]$, the running max stays at $1$. At $x=\pi$, both pieces meet smoothly: $f(\pi^-)=1$ and $2+\cos\pi=1$, and the derivatives also match ($\sin'(\pi/2)=0$ matches the flat piece, and the flat piece's derivative $0$ matches $-\sin\pi=0$ from the cosine piece at the join). So $f$ is differentiable everywhere in $(0,\infty)$.

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