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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)sint 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
(a) `because e^(x) in (1,e) " in " (0,1) and int_(0)^(x)f(t) sin t dt in (0, 1) " in " (0, 1)`
`therefore e^(x) - int_(0)^(x) f(t) sin t dt ` cannot be zero.
So, option (a) is incorrect.
(b) `f(x) + int_(0)^((pi)/(2)) f(t) sin t dt ` always positive
`therefore` 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) cost dt lt 0 rArr h(1) =1 int_(0)^((pi)/(2)-1)f(t)cost dt gt 0`
`therefore` Option (c) is correct.
(d) Let `g(x) =x^(9)-f(x) rArr g(0) = -f(0) lt 0`
g(1) =1-f(1) gt 0`
` therefore ` Option (d) is correct.
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