Show that $f$ is injective in $(a,b)$

Question

The problem that I have trouble with says:

Suppose that $f$ has second derivative in $(a,b)$ show that $f$ is injective in $(a,b)$

-Can you help me please how can I proceed? The only things that I know but I don't know how to write or demonstrate is the definition of injective function and I don't know if I need to apply the second derivative criterion and the mean value theorem

Please, any hints?

Answer 1

Note that every polynomial has second derivative.

Not every polynomial is injective.

For example $$f(x)=x^2 +1$$ is not injective because $$f(-1)=f(1)=2$$ but it has second derivative. $$ f''(x) = 2$$

Thus the statement of the problem is not true.