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y=f(x) satisfies the relation int(2)^(x)...

`y=f(x)` satisfies the relation `int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt`
The range of `y=f(x)` is

A

`[0,oo)`

B

`R`

C

`(-oo,0]`

D

`[-1/2,1/2]`

Text Solution

Verified by Experts

The correct Answer is:
D
`int_(2)^(x)f(t)dt=(x^(2))/2+int_(x)^(2)t^(2)f(t)dt`
Differentiating w.r.t `x` we get

`f(x)=x+(-x^(2)f(x))`
or `f(x)[1+x^(2)]=x`
Let `y=f(x)=x/(1+x^(2))`
or `yx^(2)-x+y=0`
Since `x` is real `Dge0`
or `1-4y^(2)ge0`
or `y epsilon [-1/2,1/2]`
Also `f(x)` is an odd funtion. Hence `int_(-2)^(2)f(x)dx=0`
`f'(x)=(1+x^(2)-2x^(2))/(1+x^(2))-(1-x^(2))/(1+x^(2))ge0`
or `x^(2)-1le0`
or `x epsilon[-1,1]`
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CENGAGE-DEFINITE INTEGRATION -Exercise (Comprehension)
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  2. y=f(x) satisfies the relation int(2)^(x)f(t)dt=(x^(2))/2+int(x)^(2)t^(...

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  3. y=f(x) satisfies the relation int(2)^(x)f(t)dt=(x^(2))/2+int(x)^(2)t^(...

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  4. Let f:RtoR be a differentiable function such that f(x)=x^(2)+int(0)^(x...

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  5. Let f:RtoR be a differentiable function such that f(x)=x^(2)+int(0)^(x...

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  6. Let f(x) be a differentiable function such that f(x)=x^2 +int0^x e^-t ...

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  7. f(x) satisfies the relation f(x)-lambdaint0^(pi//2)sinx*costf(t)dt=si...

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  8. f(x) satisfies the relation f(x)-lamda int(0)^(pi//2)sinxcostf(t)dt=si...

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  9. f(x) satisfies the relation f(x)-lamda int(0)^(pi//2)sinxcostf(t)dt=si...

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  10. Check whether the equation is quadratic equation: (x-1) (x-5)=(x-1)(x-...

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  11. Let f(x) and phi(x) are two continuous function on R satisfying phi(x)...

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  12. If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and pos...

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  13. Evaluate int(0)^(2)(x^2+x+2)dx

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  14. Evaluating integrals dependent on a parameter: Differentiate I with ...

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  15. Evaluate int(2)^(3)(x^2+1)dx

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  16. Evaluating integrals dependent on a parameter: Differentiate I with ...

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  17. Evaluating integrals dependent on a parameter: Differentiate I with ...

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  18. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt The range of f(x) is

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  19. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt f(x) is not invertibl...

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  20. f(x)=sinx+int(-pi//2)^(pi//2)(sinx+tcosx)f(t)dt The value of int(0)^(...

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