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Is there a function $f$ such that $f(x)<0, f'(x)<0,f''(x)>0 \,\forall x\in \mathbb R$ ?

I think there is, since it would be quite strange that there is not (and maybe hard to prove). After trying some logarithms, exponentials and polynomials I am out of ideas.

Definitely polynomials doesn't work: If $f(x)$ is a polynomial such that $f(x)<0 \forall x\mathbb \in \mathbb R$ then it is clear that the leading exponent must be even and the coefficient of it has to be negative. But then after derivating we violate our initial condition for $f'(x)$. Any ideas?

Note: Given the level level of the other questions in the book, the most likely functions that a simple example would use are logarithms and exponentials, trigonometric functions and their inverses, and polynomials. However, any example is welcome.

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There is no such function. Since $f''>0$ it is strictly convex, and its graph is above the tangent at any point. Then $$ f(x)\ge f(0)+f'(0)\,x\quad\forall x\in\mathbb{R}. $$ Since $f'(0)<0$, $f(x)>0$ if $x<-|f(0)/f'(0)|$.

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