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Question : 28
Total: 36
If y = s i n − 1 (
) , then show that
=
OR
Verify the Rolle's Theorem for the functionf ( x ) = e x cos x in [ −
,
]
OR
Verify the Rolle's Theorem for the function
Solution: 👈: Video Solution
Put x = cos 2 θ
⇒ y = s i n − 1
(
+
)
⇒ y = s i n − 1
(
+
)
⇒ y = s i n − 1 (
+
)
⇒ y = s i n − 1 ( s i n (
+ 2 θ ) .
⇒ y =
+ 2 θ .
⇒
= 2
Put θ =
⇒
=
∴
=
OR
As we know that exponential and cosine functions are continuous and differentiable onR .
Let us find the values of the function at an extreme
⇒ f ( −
) = e −
cos ( −
)
⇒ f ( −
) = e −
× 0
⇒ f ( −
) = 0
⇒ f (
) = e
cos (
)
⇒ f ( π ) = e
× 0
⇒ f ( π ) = 0
Here,f ′ ( − π ∕ 2 ) = f ( π ∕ 2 ) , therefore there exist a c ∈ ( − π ∕ 2 , π ∕ 2 ) such that f ′ ( c ) = 0 .
Let us find the derivative off ( x )
⇒ f ′ ( x ) =
⇒ f ′ ( x ) = cos x
+ e x
⇒ f ′ ( x ) = e x ( − s i n x + cos x )
Here, f ′ ( c ) = 0
⇒ e c ( − s i n c + cos c ) = 0
⇒ − s i n c + cos c = 0
⇒
s i n c +
cos c = 0
⇒ − s i n (
) s i n c + cos (
) cos c = 0
⇒ cos ( c +
) = 0
⇒ c +
=
⇒ c =
E ( −
,
)
Thus, Rolle's theorem is verified.
OR
As we know that exponential and cosine functions are continuous and differentiable on
Let us find the values of the function at an extreme
Here,
Let us find the derivative of
Thus, Rolle's theorem is verified.
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