AP EAPCET 24-Aug-2021 Shift 1 Paper

f(x)=xa‌log‌x and f(0)=0
∵ Rolle's theorem is applicable in [0,1]
⇒ It is continuous in closed interval [0,1]
and It is differentiable in (a,b) and f(0)=f(1),
∵ f(x) is continuous at x=0
∵

lim

x→0+

f(x)=f(0)
⇒

lim

x→0+

xa‌log‌x=0
⇒

lim

x→0+

log‌x

x−4

=0 [∵ It is −

∞

∞

indeterminate form]
Using L' hospital rule

lim

x→0+

1 x

−a.x−a−1

=0
⇒

lim

x→0+

−

xa

a

=0
∵ It is possible when a>0⇒a=

1

2