İ want just a simple explaining about the derivatives with the nondecreasing functions : İf we have a nondecreasing function f and :
$ {f}\prime{(}{t}_{1}{)}\leq{f}\prime{(}{t}_{2}{)} $ Does it lead to :
$ {t}_{1}\mathrm{\leq}{t}_{2} $ İs the converse true ?? İ know that if the function nondecreasing and we have :
$ {t}_{1}\mathrm{\leq}{t}_{2} $ Then we
$ {f}{\mathrm{(}}{t}_{1}{\mathrm{)}}\mathrm{\leq}{f}{\mathrm{(}}{t}_{2}{\mathrm{)}} $ Thanks in advance
Consider $f(x)=x^3$. Then, $f'(x)=3x^2$ and we have $f'(0)\leq f'(-1)$ as $0\leq 3$. This shows that both implications do not hold.