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- Homework Statement:
- show ##f(x)\in C^2(\mathbb{R}##

- Relevant Equations:
- $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$

this seems to come up frequently in undergrad math classes so it is worth asking, what is the simplest and most efficient way to show ##f(x)\in C^2(\mathbb{R})##

given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$

And what is the most through and concrete way to show it?

one could compute ##f(x)##, ##f'(x)## and ##f''(x)## at ##x=-\frac{1}{2}## to show whether they exist and are continuous. and polynomial functions are ##\in C^\infty## so we only have to consider the boundary. is this sufficient?

given $$f(x)=\begin{cases} (x+1)^4 & x<-\frac{1}{2} \\ 2x^4-\frac{3}{2}x^2+\frac{5}{16} & -\frac{1}{2}\leq x \end{cases}$$

And what is the most through and concrete way to show it?

one could compute ##f(x)##, ##f'(x)## and ##f''(x)## at ##x=-\frac{1}{2}## to show whether they exist and are continuous. and polynomial functions are ##\in C^\infty## so we only have to consider the boundary. is this sufficient?

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