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Question

Let f:R(0,1) be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

A
xπ2x0f(t)cost dt
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B
x9f(x)
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C
exx0f(t)sint dt
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D
f(x)+π20f(t)sint dt
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Solution

The correct option is B x9f(x)
A)
As range of f(x) is (0,1),
Let g(x)=x9f(x)
g(x)=0x=(f(x))19
x will lie (0,1).

B)
Let g(x)=xπ2x0f(t)cost dt
g(0)=π20f(t)cost dt g(0)<0
Now,
g(1)=1π210f(t)cost dt
The minimum value of g(1) takes place when π210f(t)cost dt is maximum
Max.π210f(t)cost dt=π21
Min.(g(1))=2π2>0
g(1)>0
As g(0)g(1)<0 there is atleast one value of x(0,1) where g(x) will be zero.

C)
Let g(x)=exx0f(t)sint dt
g(x)=exf(x)sinx
When x(0,1) then ex(1,e) and (f(x)sinx)(0,1)
g(x)>0 x(0,1)
So g(x) monotonically increasing function.
Now, g(0)=1
So g(x)>1 x(0,1)

D)
It will always remain positive in (0,1)

Hence option A and B are correct.

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