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Let f : R to R be a continuous function ...

Let f : R `to` R be a continuous function satisfying
`f(x)+underset(0)overset(x)(f)"tf"(t)"dt"+x^(2)=0`
for all `"x"inR`. Then-

A

`lim._(xto-oo)f(x)=2`

B

`lim._(xto-oo)f(x)=-2`

C

f(x) has more than one point in common with x-axis

D

f(x) is an odd functions

Text Solution

Verified by Experts

The correct Answer is:
B
`f(x)+f_(0)^(x)tf(t)dt+x^(2)=0`
`f^(')(x)+xf(x)+2x=0`
`(f^(')(x))/(f(x)+2)=-x`
`"f"(f^(')(x))/((f(x)+2))dx=-fxdx`
`ln(f(x)+2)=-(x^(2))/(2)+c`
`f(x)+2=e^(-x^(2)//(2))+c`
`f(x)=ke^(-x^(2)//2)-2`
where x=0 f(x)=0 k=0
f(x)=2 `(e^(-x^(2//2))-1)`
clearly `underset(xto-oo)lim f(x)=-2`
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Knowledge Check

  • Let f: R rarr R be a continous function satisfying f(x) = x + int_(0)^(x) f(t) dt , for all x in R . Then the numbr of elements in the set S = {x in R: f (x) = 0} is -

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  • Let S be the set of real numbers p such that there is no nonzero continuous function f: R to R satisfying int_(0)^(x) f(t) dt= p f(x) for all x in R . Then S is

    A
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    the set of all rational numbers
    C
    the set of all irrational numbers
    D
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