BITSAT 2012 Paper

For Rolle’s theorem in [a, b], f(a) = f(b),
In [0, 1] ⇒ f(0) = f(1) = 0
Since the function has to be continuous in [0, 1]
⇒ f (0) =

lim

x→0+

f (x) = 0
⇒

lim

x→0−

xα log x = 0 ⇒

lim

x→0

logx

x−α

= 0
Applying L.H. Rule

lim

x→0

1∕x

−αx−α−1

= 0
⇒

lim

x→0

−xα

α

= 0 ⇒ α > 0