Exercise 5.2.10

Recall that a function $f:(a,b)\to R$ is increasing on $(a,b)$ if $f(x)\le f(y)$ whenever $x<y$ in $(a,b)$. A familiar mantra from calculus is that a differentiable function is increasing if its derivative is positive, but this statement requires some sharpening in order to be completely accurate. Show that the function

is differentiable on $R$ and satisfies $g^{\prime }(0)>0$. Now, prove that $g$ is not increasing over any open interval containing 0 .

In the next section we will see that $f$ is indeed increasing on $(a,b)$ if and only if $f^{\prime }(x)\ge 0$ for all $x\in (a,b)$.