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Suppose that $f$ is continuous and positive on $(0, 1)$ and that $\lim_{x \to 0^+} f(x) = \lim_{x \to 1^−} f(x) = 0.$ Show that $f$ has a maximum value on $(0, 1)$ but no minimum value.

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HINT: Define $g:[0,1]\to\Bbb R:x\mapsto\begin{cases}0,&\text{if }x=0\text{ or }x=1\\f(x),&\text{if }0

  1. Show that $g$ is continuous and therefore has both a maximum and a minimum on $[0,1]$.

  2. Show that $g$ actually has a maximum on $(0,1)$, which is therefore also a maximum of $f$.

  3. Show that for any $a>0$ there is an $x\in(0,1)$ such that $f(x); conclude that $f$ has no minimum.