Let f be 2 times differentiable function in open interval (0,1) with following conditions:
1. lim(x-->0)f(x) = lim(x-->1)f(x) = 0
2.Exist M>0 so that |f ''(x)|<=M for all x in (0,1)
Prove that |f '(x)|<=M/2
Thank you all!
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 .