14 lines
349 B
TeX
14 lines
349 B
TeX
\beginsection 18.7
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Prove that $x2^x=1$ for some $x$ in $(0,1)$.
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\medskip
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We define the function $f(x)=x2^x$.
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For this function we have $f(0)=0$ and $f(1)=2$.
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Since $f(0)<1<f(1)$, by IVT $f(x)=1$ must occur somewhere in the
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closed interval $[0,1]$.
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We have $f(0)\ne1$ and $f(1)\ne1$ so $x2^x=1$ must occur somewhere in the
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open interval $(0,1)$.
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