24 lines
908 B
TeX
24 lines
908 B
TeX
\beginsection 18.5
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(a) Let $f$ and $g$ be continuous functions on $[a,b]$ such that
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$f(a)\ge g(a)$ and $f(b)\le g(b)$. Prove that $f(x_0)=g(x_0)$ for at least
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one $x_0$ in $[a,b]$.
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\medskip
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Define a function $h=f-g.$
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The function $h$ is continuous because $f$ and $g$ are continuous
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(theorem 17.4, p. 92).
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We have $h(a)\ge0$ and $h(b)\le0$.
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Now we can apply IVT and assert that some $x_0$ exists at which $h(x_0)=0$
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(simply replace $y$ with 0 in the theorem).
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Since $h(x_0)=f(x_0)-g(x_0)=0$, we have proved that $f(x_0)=g(x_0)$.
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\medskip
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(b) Show that Example 1 can be viewed as a special case of part (a).
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\medskip
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Example 1 includes a ``little trick'' if defining $g(x)=f(x)-x$.
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(There's the hint for the solution of 18.5 (a) above.)
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The $x$ can be considered a function so we have the difference of two
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functions as in part (a). The example is a special case because we have the
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specific function $x$.
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