If $f(x) >0$ $\forall x,f(0)=1,f'(0)=-1$ then is $f''(x)>0$ for all $x$?

Question

If $f(x)>0$ $ \forall x$ and $f(0)=1, f'(0)=-1$
then is it true to say $f''(x)>0$ for all values of $x$?

I tried the following example: Let $f(x)=\exp(-x)$, then
$$f(0)=\exp(0)=1,\ \ f'(x)=- \exp(-x) \implies f'(0)=-1.$$ So we find $f''(x)= \exp(-x) > 0$ for all $x$.

So the above statement is true at least for this $f$.

But I don't know whether the statement is true for any $f(x)$?

Any help will be appreciated.

Answer 1

The function $$f(x)=2-\cos(x)-\sin(x) $$ is always positive, and satisfies $f(0)=1,f'(0)=-1$. However, $f''(x) >0$ does not always hold. In other words, your proposition is false.