If f(x)= array c x ( 1 x ), x 0 , then at x=0 the 0, x=0 array . function f(x) is

Step-by-step solution

Given, f(x)= array c x ( 1 x ), x 0 0, x=0 array . For continuity at x=0 , LHL =f(0-0)= _ h 0 f(0-h) aligned &= _ h 0 (-h) (- 1 h ) &=0 ( finite quantity ) &=0 RHL &=f(0+0)= _ h 0 f(0+h) &= _ h 0 (h) ( 1 h ) aligned =0 ( finite quantity )=0 and f(0)=0 f(0)= LHL = RHL f(x) is continuous at x=0 . For differentiability at x=0 , R f^ (0)= _ h 0 f(0+h)-f(0) h aligned &= _ h 0 h ( 1 h )-0 h &= _ h 0 ( 1 h ) &=( a finite quantity persist between -1 & & to +1) &= does not exist L f^ (0) &= _ h 0 f(0-h)-f(0) -h &= _ h 0 (-h