13 lines
481 B
TeX
13 lines
481 B
TeX
\beginsection 29.11
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Show that $\sin x\le x$ for all $x\ge0$.
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{\it Hint:} Show that $f(x)=x-\sin x$ is increasing on $[0,\infty).$
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\medskip
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$$f^\prime(x)=1+\cos x$$
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Since the range of $\cos x$ is $[-1,1]$, the range of $f^\prime(x)$ is
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$[0,2]$. By corollary 29.7 we conclude that $f$ is an increasing function
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because $f^\prime(x)\ge0$ for all $x\in[0,\infty)$.
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Since $f(0)=0$ and $f$ is an increasing function we must have
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$f(x)\ge0$ for $x\in[0,\infty)$.
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Therefore $x\ge\sin x$. |