Construct a real function with is exactly C^2 such that its first derivative does not vanish everywhere

Question

I need to construct a real function with is exactly C^2 (that is, it is continuous and 2 times differentiable but it is not 3 times differentiable) such that its first derivative never vanishes.

I tried

x^5 Sin(1/x) + Exp(x)

x^(5/2) + Exp(x)

|x|^3 + Exp(x)

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

Answer 1

$$f(x) = \begin{cases}x^2+2x & x\ge 0 \\ -x^2+2x & x<0\end{cases}$$

$$f'(x) = \begin{cases}2x+2 & x\ge 0 \\ -2x+2 & x<0\end{cases}$$

$$f''(x) = \begin{cases}2 & x\ge 0 \\ -2 & x<0\end{cases}$$

And $f'''$ is not differentiable at $0$.