I have a question regarding the derivative of two functions.
I am wondering if the following statement is true: If the derivative of a function (say, $f_1$) is smaller than that of the other function (say, $f_2$), then $f_1$ < $f_2$.
Mathematically speaking,
If $\frac{d}{dx} f_1(x) < \frac{d}{dx} f_2(x)$ for all $x$, then is it true that $f_1 < f_2$ for all $x$?
Thank you in advance!!
Best Answer
No. Take $$ f_1(x) = e^{-x}, \quad f_2(x) = 0. $$ Then $$f_1'(x) = -e^{-x} < 0 = f_2'(x) $$ while $$ f_1(x) = e^{-x} > 0 = f_2(x). $$