Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a) < g(a)$ and $f(b) > g(b)$

Question

Suppose $f$ and $g$ are continuous on $[a,b]$ and that $f(a) < g(a)$ but $f(b) > g(b)$. Prove that $f(x) = g(x)$ for some $x \in [a,b]$.

I did try and actually did get as far as coming up with $f-g$. Would I set $h(x)= f(x)- g(x)$ and say $h(a) <x< h(b)$?

I did try and actually did get as far as coming up with f-g. Would i set h(x)= f(x)- G(x) and say H(a) <x< H(b)? — John Snow, Apr 01 '13 at 18:54
For $H=f-g$, you have $H(a)<0$ and $H(b)>0$. Then $H(a)<0<H(b)$. Since $H$ is continuous ... — David Mitra, Apr 01 '13 at 19:01

score 14 · Accepted Answer · answered Apr 01 '13 at 18:44

14

Hint: Apply the Intermediate Value Theorem to the function $f-g$ on the interval $[a,b]$.

answered Apr 01 '13 at 18:44

David Mitra

1 Answers1