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I need to construct a real function with is exactly $C^2$ (that is, it is continuous and two times differentiable but it is not three times differentiable) such that its first derivative never vanishes.

I tried

$x^5 \sin(\frac{1}{x}) + \exp(x)$

$x^{\frac{5}{2}} + \exp(x)$

${|x|}^3 + \exp(x)$

all these functions are exactly $C^2$, but their first derivatives vanish in some point(s). :(

Oussama Boussif
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JI-br
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    "not three dimes differentiable" ....in at least one point of its original domain. Is this what you meant? And hwta you wrote in the title doesn't seem to be what you want: "does not vanish everywhere" could easily be understood as "it doesn't vanish in at least one point" . Finally, what domain do you want for your function? Or it doesn't matter? – DonAntonio May 17 '17 at 17:04
  • Yes, I mean "not three dimes differentiable in at least one point of its original domain".

    As for "does not vanish everywhere" I mean that if f:U->R is the function that I am looking for, with U an open connected set of R, then either df(x)/dx >0 or df(x)/dx <0 or all x in U.

     – JI-br May 17 '17 at 23:10

1 Answers1

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Define $$g= \left \{ \begin{matrix} x^4& x>0\\ x^3& x\leq0 \end{matrix} \right. $$ Now $g$ satisfies all the conditions except that its first derivative vanishes at the origin. So, set $\tilde {g}=g+10x$.

clark
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