As a simple example, one can look at f(x)=1/x in R+. The derivative is f'(x)=-1/x^2 and the second derivative is f''(x)=2/x^3. In general, every function of the form f(x)=1/x^n with odd n will have this behaviour, and if n is even the behaviour is the opposite. And yes, many more examples can be found.
More generally, this is why I'm wary of indirect indicators. They never tell the whole story, and because of that they're used disingenuously in order to muddy the waters. You see this a lot in big scale PR campaings, such as climate change denialism, and pro-sugar and pro-alcohol disinformation (we had a lot of pro-tobacco as well, but it has subsided in the last two decades or so, at least in the West).
For those who find the existence of such analytic functions intuitively "wrong", note that it is essential to this example that though the second derivative is positive, it is positive and decreasing towards 0. If the second derivative is bounded from below by some positive value, then eventually the first derivative will become positive and it will diverge towards positive infinity and thus the function itself will diverge towards positive infinity.
More precisely suppose that f is twice continuously differentiable and f'(x) < 0 and f''(x) ≥ 0 for all x greater than some K (for example K=0 in the examples given) then λ( (f'')^-1((a, ∞)) ∩ (K, ∞) ) < ∞ for all a > 0 where λ is the Lebesque measure.
Yes exactly. There are folks who would ride that positive second derivative all the way down to f(x) = 0 and believe every step of the way that the indirect indicator is telling them that we're going to turn around any second now.
A good example for pundit disproving is f(t) = exp(-t), the solution to the differential equation dx/dt = -x. So the continuous limit of something like "every year, 1% of the ice caps melt." Look, the volume of ice cap lost is decreasing year over year!
More generally, this is why I'm wary of indirect indicators. They never tell the whole story, and because of that they're used disingenuously in order to muddy the waters. You see this a lot in big scale PR campaings, such as climate change denialism, and pro-sugar and pro-alcohol disinformation (we had a lot of pro-tobacco as well, but it has subsided in the last two decades or so, at least in the West).