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If I= int(0)^(1)(x)/(8+x^(3))dx then the...

If `I= int_(0)^(1)(x)/(8+x^(3))dx` then the smallest interval is which I less is

A

(0,1/8)

B

(0,1/9)

C

(0,1/10)

D

(0,1/7)

Text Solution

Verified by Experts

The correct Answer is:
B
Let `f(x) =(x)/(8+x^(3))`. Then
`f'(x)=(8+x^(3)-3x^(3))/(8+x^(3))=(2(4-x^(3)))/((8+x^(3))^(2))gt 0 "for all" x in [0,1]`
`rArr f(x)` is increasing on [0,1]
`rArr f(0) le f(x) le f(x) le f(1) "for all" x in [0,1]`
`rArr 0 lt f(x) le (1)/(9)`
`rArrr (1-0)xx0 lt underset(0)overset(1)int f(x)dxlt (1)/(9)(1-0)`
`rArr 0lt underset(0)overset(1)int (x)/(8+x^(3)) dxlt (1)/(9) rArr I in (0,1//9)`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
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  11. Let f: R in R be given by f(x)={{:(|x-[x]|,"when[x]is odd"),(|x-[x]-...

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  12. If f(x) = sin x +cos x and g(x) = {:{((|x|)/(x),","x ne0),(2,","x=0):}...

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  13. If x in[(4n+1)(pi)/(2),(4n+3)(pi)/(2)] and n in N, then the value of i...

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  14. If f:R in R is continuous and differentiable function such that int(...

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  18. If f:R in R be such that f(x)=sqrt(sin(cosx))+"In"(-2cos^(2) x+3 cos...

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