## About a bound of a function. [closed]

Define $f : [0,\infty) \to \Bbb R$ as $f(x) := \frac{1}{10} (1+xe^x) -x$. And let $x_0$ be a first zero of $f$, i.e., $f(x_0) = 0$. Let $g$ be a nonnegative function defined on $[0,a]$. Then if $g(0) < x_0$ and $f(g(x)) >0$ for all $x \in [0,a]$, then how can I conclude that $g(x) < x_0$ for all $x \in [0,a]$?

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