Below is the graph of the derivative of a function $f$ which is continuous on all real numbers
Which one of the followings are always correct?
I. $f(2)\lt f(4).$
II. The function $f$ has a local minimum at $x=3$.
III. The function $f$ is increasing on the interval $(-\infty,-1)$
I can figure out the last two options:
Firstly, third option is wrong because we see that the derivative of this function changes from negative to positive at a point smaller than $-1$, so the function is not always increasing on $(-\infty,-1)$.
Second option is correct because the derivative changes from negative to positive at the point $x=3$, meaning $f$ has a minimum point at 3.
Now for the first option, I see that the function is decreasing at $2$ and increasing on $4$, but I believe this does not necessarily mean that $f(2)\lt f(4)$.
How do we explain that $f(2)\lt f(4)$ will always be true?