VITEEE 2015 Paper

Given: f(t) =

sint

t

At t = 0, we will check continuity of the function.
LHL = f(0 - h)
=

lim

h→0

sin(0−h)

(0−h)

=

lim

h→0

−sinh

−h

= 1
RHL = f(0 + h)

lim

h→0

sin(0+h)

(0+h)

=

lim

h→0

sinh

h

= 1
and f(0) = 1
LHL = RHL = f(0)
So, the function is continuous at t = 0
Now, we check the function is maximum or minimum.
f ' (t) =

1

t

cos t -

1

t2

sin t
and f " (t) =

−1

t

sin t -

1

t2

cos t -

1

t2

cost +

2

t3

sin t
=

−sint

t

-

2cost

t2

+

2sint

t3

For maximum or minimum value of f(x), put
⇒

cost

t

f (x) = 0
-

sint

t

= 0 ⇒

tant

t

= 1
Now , li

m

t→0

f " (t)
= - li

m

t→0

(

sint

t

) - 2 li

m

t→0

(

tcost−sint

t3

) [0/0 form]
= - 1 - 2 li

m

t→0

(

cost−tsint−cost

3t2

)
[using L’ Hospital rule]
= - 1 +

2

3

li

m

t→0

sint

t

= - 1 +

2

3

× 1 =

−1

3

< 0
So, function f(t) is maximum at t = 0