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If $f$ and $g$ are two continuous functions from $[a,b]$ to real numbers s.t $f(a)\lt g(a)$ and $f(b)\gt g(b)$. Prove that $f(x)=g(x)$ has at least 1 solution $c$ in $(a,b)$.

I tried using intermediate value theorem. It states that if f continuous on [a,b] f(a)< d< f(b) then there exists a point c in (a,b) such that f(c)=d

vonbrand
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sarah
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1 Answers1

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For part $(a)$, let $h(x)=f(x)-g(x)$. Notice the conditions imply that $h(a)<0$ and $h(b)>0$. Since $h$ is continuous, the intermediate value theorem gives us some $c$ in $(a,b)$ where $h(c)=0$. See if you can finish the rest!

Jared
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