1.

which of the following is true for x in [0,1]?

Answer»

`sin^(-1)x+x^(2)-x(9-x^(2))/(3)le0`
`sin^(-1)x+x^(2)-x(9-x^(2))/(3)ge0`
`sin^(-1)x+x^(2)-x(9-x^(2))/(3)le0`
`sin^(-1)x+x^(2)-x(9-x^(2))/(3)ge0`

Solution :Let `f(x)=sin^(-1)x+x^(2)-3x+(x^(3))/(3)`
`THEREFORE f(X)=(1)/sqrt(1-x^(2))+2x-3+x^(2)`
Thus f(X)=0 for some x=`x_(1) in (0,1)`

`f(x)=(x)/(1-x^(2))^(3//2)+2+2xgt 0 forall x in (0,1)`
Thus x=`x_(1)`is the point of MINIMUM
f(0)=0,f(1)=`pi//2-5//3lt0`
f(X) is global maxima `forall x in [0,1]`.Thus
`f(X)lef(X)XIN [0,1]or sin ^(-1)x+x^(2)-3x+x^(3)//3le0`
or `sin^(-1)x+x^(2)LEX(9-x^(2))/(3)forall x in [0,1]`


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