Let $f$:$[a,b](0,+)g$:$[a,b]Rg(x)={\displaystyle \underset{a}{\overset{x}{}}}f(t)dt+{\displaystyle \underset{b}{\overset{x}{}}}\frac{1}{f(t)}$dtga,bga,$b5\mathrm{.}4ga,babe$ real numbers, let $f$:$[a,b](0,+)be$ a positive continuous function, and let $g$:$[a,b]Rbe$ the function

$g(x)={\displaystyle \underset{a}{\overset{x}{}}}f(t)dt+{\displaystyle \underset{b}{\overset{x}{}}}\frac{1}{f(t)}dt$

$(a)$ Show from definition $(Definition1\mathrm{.}54)$ that $gis$ strictly increasing $ona,b.$

$(b)$ Suppose $itis$ given that $gis$ continuous $ona,b(This$ fact actually follows immediately from Lemma $5\mathrm{.}53$ later $in5\mathrm{.}4).$ Show that $g$ has one and only one root $ina,b.$