In which of the following function is rolle’s theorem applicable?

detailed solution

Correct option is D

A) Discontinuous at x=1 ⇒ not applicable. B) F(x)  is not continuous (jump discontinuity) at x=0.C) Discontinuity (missing point) at x=1 ⇒  not applicable.D) notice that x3−2x2−5x+6=(x−1)(x2−x−6). hence, f(x)=x2−x−6 if x≠1 and f(1)=−6. ⇒f  is continuous at x=1. So f(x)=x2−x−6  throughout the interval [−2,3]. Also, note that f(−2)=f(3)=0.  hence, Rolle’s theorem applies. f'(x)=2x−1. Setting f'(x)=0,  we obtain x= 12  which lies between –2 and 3.