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Let f: R rarr (0,1) be a continuous func...

Let `f: R rarr (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

`e^(x)-int_(0)^(x)f(t) sin " t dt"`

B

`f(x)+int_(0)^(pi/(2))f(t)sin "t dt"`

C

`x-int_(0)^(pi/(2)-x)f(t)cos " t dt "`

D

`x^(9)-f(x)`

Text Solution

Verified by Experts

The correct Answer is:
C, D
`:'e^(x) in (1, e) " in " ( 0 , 1) and int_(0)^(x)f(t)sin "t dt " in (0, 1 )`
`:. e^(x)- int_(0)^(x) f (t) sin " t dit "` cannot be zero.
So , option (a) is incorrect.
(b) `f(x) +int_(0)^(pi/(2)) f (t) sin ` t dt always positive
`:.` Option (b) is incorrect.
( c) Let h (x) `=x-int_(0)^(pi/(2)-x)f(t) cos "t dt"`
`h(0)=-int_(0)^(pi/2)f( t) cos " t dt " lt 0`
`h(1)=1-int_(0)^(pi/2-1)f(t) cos " t dt "lt 0`
`:.` Option ( c) is correct.
(d) Let g (x) `=x^(9)-f(x)`
`g(0)=-f(0)lt0`
`g(1)=1-f(1)lt0`
`:.` Option (d) is correct .
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