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f:[0,1)toR be a non increasing function ...

`f:[0,1)toR` be a non increasing function then for `alphaepsilon (0,1)`

A

`alpha int_(0)^(1)f(x) dx le int_(0)^(alpha)f(x)dx`

B

`alpha int_(0)^(1)f(x)dx ge int_(0)^(alpha)f(x)dx`

C

`alpha^(2) int_(0)^(1)f(x)dxleint_(0)^(alpha) f(x)dx`

D

`sqrt(alpha) int_(0)^(1)f(x)dxgeint_(0)^(alpha)f(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
A, C
`int_(0)^(alpha) f(x)dx`………..1
Put `x=alphat`
`:. int_(0)^(alpha) f(x)dx=alpha int_(0)^(1)f (alpha t)dt ge alpha int_(0)^(1)f(t)dt` as `alpha epsilon(0,1)`
Also `alpha int_(0)^(1) f(t)dt ge alpha^(2) int_(0)^(1)f(t)dt`
`:. int_(0)^(alpha)f(x)dxgealpha^(2)int_(0)^(1)f(x)dx`
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