A continuous and derivable decreasing function s.t. the derivative is $0$ for $t>t_0$?

Question

The question may be silly, but I am stucked with this thought...

I ask for a nonconstant continuous and derivable nondecreasing function s.t. the derivative approaches $0$ for $t\to\infty$, but more than that I might have $f'(t_0)=0$ for some $t_0$ and also for all $t>t_0$.

I really cannot think this example. For instance, $f(t)=\text{log}(t)$ does not mind, because $f'$ approaches to $0$, but $f'(t)\neq0\forall t$. And so on.

But because this question?

Because I have an equation on a model of form $\dfrac{d f}{d t}=g(t)$ with $g(t)$ nonpositive and I might have $g(t)=0$ for all $t>t_0$ for some $t_0>0$, but I cannot realize this.

Thank you so much.

Answer 1

Let $f(x)=-x^2$ when $x\le 0$ and $f(x)=0$, when $x>0$.

Answer 2

Consider the function $$f(x) = \int_{1}^{x}\frac{sin^{2}t}{t}dt$$ Since integral measures area below the graph and since the function being integrated is non-negative $f(x)$ is increasing. Of course, derivative of $f(x)$ is $\frac{sin^{2}x}{x}$ and it non-negative and is zero at $k*\pi$.

Answer 3

The constant function f(x)=2 (say) satisfies all your requirements.