JEE Advanced | 2014Set, Relation and FunctionMedium
Question
Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1.
Let F(x) =
f(√t)dt for x ∈ [0, 2]. If F′(x) = f′(x) for all x ∈ (0, 2), then F(2) equals
Let F(x) =
f(√t)dt for x ∈ [0, 2]. If F′(x) = f′(x) for all x ∈ (0, 2), then F(2) equalsOptions
A.e2 - 1
B.e4 - 1
C.e - 1
D.e4
Solution
F′(x) = f(x).2x = f′(x)
⇒
= 2x
⇒ f(x) = kex2
Given f(0) = 1 ⇒ f(x) = ex2
So F(x) =
exdx
= ex2 - 1
So F(2) = e4 - 1
⇒
= 2x⇒ f(x) = kex2
Given f(0) = 1 ⇒ f(x) = ex2
So F(x) =
exdx = ex2 - 1
So F(2) = e4 - 1
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