Use the Taylor series with remainder , where lies between x and x+t. This implies that . There are now two separate cases to consider.
If f(x) and f'(x) have the same sign, take . Then . Also, . So the inequality in the previous paragraph tells us that . Hence .
If f(x) and f'(x) have the opposite signs, take . Then . Also, . So the inequality tells us that . Hence .
The second term in that Taylor polynomial is , not .